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Modeling the bioelectronic interface in engineered tethered membranes : biosensing and the electrophysiological… Hoiles, William August 2015

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Modeling the Bioelectronic Interfacein Engineered Tethered Membranes:Biosensing and theElectrophysiological ResponsebyWilliam August HoilesB.A.Sc., The University of British Columbia, 2010M.A.Sc., Simon Fraser University, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)May 2015c© William August Hoiles 2015AbstractThe unifying theme of this thesis is the construction and predictive models of four novel teth-ered membrane measurement platforms: (i) the Ion Channel Switch (ICS) biosensor for detect-ing the presence of analyte molecules in a fluid chamber, (ii) a Pore Formation MeasurementPlatform (PFMP) for detecting the presence of pore forming proteins and peptides, (iii) aControlled Electroporation Measurement Device (CED) that provides reliable measurements ofthe electroporation phenomenon, and (iv) an Electrophysiological Response Platform (ERP) tomeasure the response of ion channels and cells to an electrical stimulus. Common to all fourmeasurement platforms is that they are comprised of an engineered tethered membrane thatis formed via a rapid solvent exchange technique developed by Dr. Bruce Cornell allowing theplatform to have a lifetime of several months. The membrane is tethered to a gold electrodebioelectronic interface that includes an ionic reservoir separating the membrane and gold sur-face, allowing the membrane to mimic the physiological response of natural cell membranes.The electrical response of the ICS, PFMP, CED, and ERP are predicted using coarse-grainedmolecular dynamics, continuum theories for electrodiffusive flow, and macroscopic fractionalorder models. Experimental measurements are used to validate the predictive accuracy of thedynamic models. These include using the PFMP for measuring the pore formation dynam-ics of the antimicrobial peptide PGLa and the protein toxin α-Hemolysin; the ICS biosensorfor measuring nano-molar concentrations of streptavidin, ferritin, thyroid stimulating hormone(TSH), and human chorionic gonadotropin (pregnancy hormone hCG); the CED for measuringelectroporation of membranes with different tethering densities, and membrane compositions;and the ERP for measuring the response of the voltage-gated NaChBac ion channel, and theresponse of skeletal myoblasts which are attractive donor cells for cardiomyoplasty. We envisagethe tethered membrane and atomistic-to-observable dynamic models presented in this thesis tobe invaluable for the future development of membrane based biosensors.iiPrefaceThe work presented in this thesis is based on the research conducted in the Statistical SignalProcessing Laboratory at the University of British Columbia (Vancouver). The research pro-gram was designed jointly based on ongoing discussion between the author and Prof. VikramKrishnamurthy. All major research work, including detailed problem specification, theoreticaldevelopments, performance analyses and identification of results was performed by the author,with assistance from Prof. Vikram Krishnamurthy. The numerical examples, simulations, anddata analyses were performed by the author. For all manuscripts the author was responsible forthe composition (e.g. figures, writing, concept formatting), numerical examples, simulations,and data analysis with frequent suggestions for improvement from Prof. Vikram Krishnamurthy.Dr. Cornell and Dr. Cranfield provided all the experimental data in this thesis. The followingprovides a detailed list of publications which are associated with the chapters presented in thisthesis.The material in Chapter 2 has appeared in the publications outlined below:Journal Article:• W. Hoiles, V. Krishnamurthy, and B. Cornell. Mathematical Models for Sensing De-vices Constructed out of Artificial Cell Membranes. Nanoscale Systems: MathematicalModelling, Theory and Applications, 1: 143–171, 2012.• W. Hoiles, V. Krishnamurthy, and B. Cornell. Membrane Bound Molecular Machines forSensing. Journal of Analytical and Bioanalytical Techniques, 7: 1-7, 2014.• W. Hoiles, V. Krishnamurthy, and B. Cornell. Modelling the Bioelectronic Interface inEngineered Tethered Membranes: From Biosensing to Electroporation. IEEE Transac-tions on Biomedical Circuits and Systems, PP(99): 1-1, 2014.The work presented in Chapter 3 has been submitted for possible publication in the journallisted below:Journal Articles:• W. Hoiles and V. Krishnamurthy. Pore Formation Dynamics of Antimicrobial PeptidePGLa in Tethered Membranes. Submitted to a peer reviewed journal, Sept. 2014.iiiPrefaceThe work presented in Chapter 4 has appeared inJournal Articles:• W. Hoiles, V. Krishnamurthy, B. Cornell, and C. Cranfield. An Engineered Membraneto Measure Electroporation: Effect of Tethers and Bioelectronic Interface. BiophysicalJournal, 107(6): 1339-1351, 2014.and has also been submitted for possible publication in the journal listed below where the authorand Dr. Gupta contributed equally to the coarse-grained molecular dynamics simulation results:Journal Articles:• W. Hoiles, R. Gupta, V. Krishnamurthy, B. Cornell, and C. Cranfield. A Cell-basedBioelectronic Interface for Controlled Electroporation Measurement: From Molecules toDevice. Submitted to a peer reviewed journal, Sept. 2014.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Ion Channel-Switch Biosensor . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Pore Formation Measurement Platform . . . . . . . . . . . . . . . . . . . 51.2.3 Controlled Electroporation Measurement Device . . . . . . . . . . . . . . 71.2.4 Electrophysiological Response Platform . . . . . . . . . . . . . . . . . . . 81.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 Biosensors and Electrophysiological Measurement . . . . . . . . . . . . . 91.3.2 Levels of Abstraction in Tethered Membrane Modeling . . . . . . . . . . 121.3.3 Ab-initio Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 131.3.4 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.5 Coarse-Grained Molecular Dynamics . . . . . . . . . . . . . . . . . . . . 131.3.6 Continuum Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.7 Reaction Rate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Ion Channel Switch Biosensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17vTable of Contents2.1.1 Main Results and Chapter Organization . . . . . . . . . . . . . . . . . . 182.2 Ion Channel-Switch Biosensor: Construction and Formation . . . . . . . . . . . 192.3 Dynamic Model of the Ion Channel-Switch Biosensor . . . . . . . . . . . . . . . 212.3.1 Fractional Order Model of the Bioelectronic Interface . . . . . . . . . . . 212.3.2 Continuum Model of the Bioelectronic Interface . . . . . . . . . . . . . . 232.3.3 Computing Model Parameters using Maximum Likelihood Estimator . . 272.4 Experimental Measurements Ion Channel-Switch Biosensor: Streptavidin, TSH,Ferritin, hCG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Experimental Setup and Numerical Methods . . . . . . . . . . . . . . . . 292.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Pore Formation Measurement Platform . . . . . . . . . . . . . . . . . . . . . . . 323.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.1 Main Results and Chapter Organization . . . . . . . . . . . . . . . . . . 343.2 Pore Formation Measurement Platform: Construction and Formation . . . . . . 363.3 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.1 Fractional Order Macroscopic Model . . . . . . . . . . . . . . . . . . . . 373.3.2 Generalized Reaction-Diffusion Continuum Model . . . . . . . . . . . . . 373.3.3 Coarse-Grained Molecular Dynamics of PGLa . . . . . . . . . . . . . . . 403.4 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 423.4.1 Diffusion of PGLa from Coarse-Grained Molecular Dynamics (CGMD) . 423.4.2 Surface Binding and Oligomerization of PGLa from Molecular Dynamics 423.4.3 Experimentally Measured Pore Formation Dynamics of PGLa . . . . . . 463.4.4 Experimental Setup, Model Validation, and Model Parameters . . . . . . 493.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 Controlled Electroporation Measurement Device . . . . . . . . . . . . . . . . . 584.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.1.1 Measurement of Electroporation . . . . . . . . . . . . . . . . . . . . . . . 584.1.2 Main Results and Organization . . . . . . . . . . . . . . . . . . . . . . . 594.2 Controlled Electroporation Measurement Device (CED): Formation and Opera-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3.1 Coarse-Grained Molecular Dynamics Model . . . . . . . . . . . . . . . . 634.3.2 Generalized Poisson-Nernst-Planck Continuum Model . . . . . . . . . . . 664.3.3 Macroscopic Fractional Order Model . . . . . . . . . . . . . . . . . . . . 694.4 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 714.4.1 Structure and Biomechanics of CED . . . . . . . . . . . . . . . . . . . . . 71viTable of Contents4.4.2 Lipid Energetics and Pore Density . . . . . . . . . . . . . . . . . . . . . . 724.4.3 Pore Conductance and Electrical Energy Required to form a Pore . . . . 744.4.4 Electrode Surface Impedance and Dynamics . . . . . . . . . . . . . . . . 774.4.5 Quality of Formed Membrane via Impedance Measurements . . . . . . . 794.4.6 Measured Dynamics of Controlled Electroporation Measurement Device . 814.4.7 Pore Population and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 824.4.8 Effect of Variations in Membrane Composition and Tether Density . . . 844.4.9 Experimental Measurements and Model Parameters . . . . . . . . . . . . 884.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925 Electrophysiological Response Platform . . . . . . . . . . . . . . . . . . . . . . . 935.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.1.1 Electrophysiological Response of the NaChBac Ion Channel . . . . . . . 935.1.2 Electrophysiological Response of Cells . . . . . . . . . . . . . . . . . . . . 945.1.3 Main Results and Organization . . . . . . . . . . . . . . . . . . . . . . . 945.2 Formation and Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.3 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.3.1 Embedded Ion Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.3.2 Cell Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.4 Numerical and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 975.4.1 Electrophysiological Response of NaChBac Ion Channels . . . . . . . . . 985.4.2 Electrophysiological Response of Cells . . . . . . . . . . . . . . . . . . . . 995.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106viiList of Tables2.1 Model Parameters for ICS Biosensor . . . . . . . . . . . . . . . . . . . . . . . . . 303.1 Diffusion Coefficients of PGLa Protomers (µm2/s) . . . . . . . . . . . . . . . . . 423.2 Model Parameter for PFMP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3 Model Parameter for α-Hemolysin (Fig. 3.10) . . . . . . . . . . . . . . . . . . . . 543.4 Model Parameter for Varying PGLa (Fig. 3.8) . . . . . . . . . . . . . . . . . . . . 543.5 Model Parameter for Varying POPG (Fig. 3.9) . . . . . . . . . . . . . . . . . . . 544.1 Diffusion Coefficient D (nm2/ns) . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.2 Parameter for Gp and Wes Predictions . . . . . . . . . . . . . . . . . . . . . . . . 914.3 Parameters for CED Current Predictions . . . . . . . . . . . . . . . . . . . . . . . 92viiiList of Figures1.1 Schematic of a natural biological membrane. The extracellular fluid representsthe content outside the cell, and the cytosol is the interior of the cell with themembrane separating the two domains. Note that biological membranes arecomposed of thousands of different components (macromolecules, lipids, chemicalspecies)–refer to [1] for a detailed description. . . . . . . . . . . . . . . . . . . . . 21.2 Overview of the engineered tethered membrane and molecular components. The“Electronics” block represents the electronic instrumentation that produces thedrive potential between the gold electrode and gold counter electrode, and recordsthe current response I(t). Go is a transient aqueous pore. The conducting gram-icidin (gA) dimer is shown and is composed of two gA monomers. A representsthe analyte species, and B the analyte receptor. MSLOH denotes synthetic ar-chaebacterial membrane-spanning lipids, DLP half-membrane-spanning tetheredlipids, DphPC and GDPE mobile half-membrane-spanning lipids, MSLB mem-brane spanning lipid, and SP a spacer. . . . . . . . . . . . . . . . . . . . . . . . 31.3 A schematic diagram of the ICS, PFMP, CED, and ERP tethered membranedevices. The tethered membrane is depicted in gray, the gold interface by thecrosshatch pattern. The unifying theme of all the devices is the use of an inertgold bioelectronic interface and an engineered tethered membrane. Sensing usingthe ICS and PFMP is performed by measuring the time-dependent impedance asa result of changes in the concentration of conducting pores. The sensing mech-anism of the CED and ERP is performed by measuring the current response ofthe devices to a time-dependent excitation potential. The current response isdependent on the concentration of conducting pores, the polarization dynam-ics of the membrane surfaces, and the charging dynamics at the surface of thebioelectronic interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Schematic diagram illustrating the length and timescale achievable by the atom-istic to macroscopic simulation methods in Sec. 1.3.2. . . . . . . . . . . . . . . . . 132.1 Fractional order macroscopic model of the ICS biosensor. The circuit parametersare defined in Sec. 2.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22ixList of Figures2.2 Schematic of the ICS biosensor model (2.9), (2.10), (2.11), and (2.12) with bound-ary conditions (2.13). The analyte enters the ICS at ∂Ωin with a flow rate of Q.∂Ωsurf is the surface of the tethered membrane, and ∂Ωb denotes the boundaryof the membrane. n denotes the inward normal vector from the surface. Otherparameters are defined in Table 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Experimentally measured and numerically predicted normalized conductanceGm(t)/Go for Streptavidin, TSH and Ferritin. The numerical predictions arecomputed using the ICS biosensor model in Sec. 2.3 with the parameters definedin Table 2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 Experimentally measured and numerically predicted normalized conductanceGm(t)/Go for the human chorionic gonadotropin (hCG) concentrations cA: 0nM, 353 nM. The numerical predictions are computed using the ICS biosensormodel in Sec. 2.3 with the parameters defined in Table 2.1. . . . . . . . . . . . . 293.1 Schematic of the Pore Formation Measurement Platform (PFMP). A voltagepotential is applied between the gold electrode and counter electrode (not shown)and the current response I(t) is measured. The PGLa peptide binds to themembrane surface, then undergoes oligomerization steps to create a PGLa porewith conductance Gp. The current response I(t) of the PFMP is dependent onthe number of conducting PGLa pores and the equilibrium number of aqueouspores present in the tethered membrane. . . . . . . . . . . . . . . . . . . . . . . . 343.2 Schematic of the atomistic-to-observable model. D is the diffusion coefficient ofbound PGLa peptides, Ĝm(t) is the predicted conductance, I(t) is the measuredcurrent from the PFMP (Fig. 3.1), and Gm(t) is the measured conductance. . . . 353.3 Schematic of the all-atom structure of PGLa (GMASKAGAIAGKIAKVALKAL-NH2) and the corresponding MARTINI coarse-grained structure constructed us-ing the protocol in [2]. The PGLa backbone beads are displayed in red, and sidechain beads in yellow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4 Snapshots of CGMD bead positions for simulation run from 0 ns to 60 ns for aDphPC membrane for the monomer binding and dimer binding of PGLa into aDphPC membrane. The NC3 bead is displayed in blue, the PO4 bead in orange,the lipid tail carbons in green beads, the PGLa backbone beads in red, and PGLaside chains using yellow beads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44xList of Figures3.5 Snapshots of the translocation of the bound monomer and dimer illustrated inFig. 3.4 into the transmembrane state via a transient hydrophilic pore. To illus-trate the structure of the hydrophilic pore both the top and side views as pro-vided, the final structure displayed is the closed aqueous pore with the monomerand dimer in the transmembrane state. The coloring scheme is identical to thatused in Fig. 3.4 with water illustrated using the translucent blue beads. . . . . . 453.6 Snapshots of the oligomerization of four PGLa monomers in the transmembranestate (Fig. 3.5). The initial position of the monomers is provided at 0 ns. At 55ns a snapshot from the side view looking into the membrane is provided. At 200ns the two formed transmembrane dimers. The coloring scheme is identical tothat used in Fig. 3.4 with water illustrated using the translucent blue beads. . . 463.7 The measured and predicted impedance (phase is represented by ∠Z(f) in de-grees and magnitude by Z(f)) of the 10% tether density DphPC bilayer mem-brane. The solid line is the predicted and the dotted the experimentally mea-sured. The grey and black colours indicate the impedance for two identicallyconstructed membranes. All predictions are computed using (3.1). The experi-mental results are extracted from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . 473.8 Experimentally measured and numerically predicted conductance for DphPCtethered membrane with 10, 20, 30, and 40 µM of PGLa. The predictions aremade using (3.4), (3.5) with the reaction mechanism given by (3.2) and simula-tion parameters provided in Table 3.4. The experimental results are extractedfrom [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.9 Experimentally measured (dotted) and numerically predicted (solid line) mem-brane conductance for tethered membranes composed of 10%, 20%, 30%, 40%,and 50% POPG with a PGLa concentration of 30 µM. The predictions are madeusing (3.4), (3.5) with the reaction mechanism given by (3.2) and simulationparameters provided in Table 3.5. The experimental results are extracted from [4]. 493.10 Experimentally measured (dotted) and numerically predicted (solid line) conduc-tance for DphPC tethered membrane and α-Hemolysin concentration of 3 µM.The predictions are made using (3.4), (3.5) with the reaction mechanism givenby (3.9) and simulation parameters provided in Table 3.3. The experimentalresults are extracted from [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51xiList of Figures3.11 Panel (a) illustrates how the impedance changes for the Cm values of 5 nF, 10nF, 15 nF, and 20 nF with all other parameters fixed. (b) illustrates how theimpedance changes for the Cdl values of 100 nF, 130 nF, 160 nF, and 190 nFwith all other parameters fixed. (c) illustrates how the impedance changes forthe Re values of 100 Ω, 500 Ω, 1000 Ω, and 1500 Ω with all other parametersfixed. The predicted impedance (phase is represented by ∠Z(f) in degrees andmagnitude by Z(f)) is computed using (3.1). . . . . . . . . . . . . . . . . . . . . 553.12 The predicted impedance (phase is represented by ∠Z(f) in degrees and mag-nitude by Z(f)) for varying membrane conductance Gm with Cm, Cdl, and Refixed. All predictions are computed using (3.1). . . . . . . . . . . . . . . . . . . . 553.13 PFMP and a schematic of the computational domain for the generalized reaction-diffusion continuum model. The parameters are defined in Table 3.2. ∂Ωb is theboundary of the tethered membrane illustrated by the black boxes, and ∂Ωin isthe analyte input flow-chamber indicted in gray. . . . . . . . . . . . . . . . . . . 564.1 Schematic of the controlled electroporation measurement device and atomistic-to-observable model. The “Electronic” block applies a potential between theelectrodes, this increases the transmembrane potential Vm, and records the cur-rent response I(t). Gp is the conductance of the aqueous pore. In the atomistic-to-observable model the coarse-grained molecular dynamics (CGMD) is used tocompute the diffusion D, thickness of the membrane hm, surface tension σ, andline tension γ. The Continuum model is used to compute the pore conductanceGp and electrical energy required to form a pore Wes. The Macroscopic modelis used to relate Gp,Wes, σ, and γ to the experimentally measured current I(t).In the CGMD panel, the yellow beads model the bioelectronic interface, thetranslucent blue beads the water, and the green beads the tethered membrane. . 624.2 Ribbon structure of 0% tethered DphPC membrane. Lipid tails are representedby the green beads, NC3 bead is displayed in blue, the PO4 bead in orange, OHbead in red, the COC bead as pink, and the water beads as a translucent blue.The coloring scheme of the axis is red for x, blue for y, and green for z. Notethat this axis is only used for computing the line tension of of the membrane asdiscussed in Sec. 4.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Pore geometry used to compute the pore conductance Gp. . . . . . . . . . . . . . 67xiiList of Figures4.4 Snapshots of restrained DphPC lipid in 25% DphPC membrane for umbrellasampling. (a) is the lipid in the equilibrium position, (b) the lipid at the centerof the bilayer, and (c) the lipid in bulk water. Water is represented by light bluebeads, pulled lipid as magenta spheres, lipid tails as green lines, DphPC andGDPE headgroups NC3,PO4,OH as blue,orange and red balls, tethers as violetsticks, and spacers as tan sticks. The gold is represented by the yellow beads. . . 734.5 (a) is the associated PMFs for a 0% and 25% DphPC lipid membrane. (b) and(c) provide the CGMD simulation snapshots of the restrained DphPC lipid atthe center of the membrane used to construct the PMF for the 0% and 25%tethered DphPC respectively. Water is represented by light blue beads, pulledlipid as magenta spheres, lipid tails as green lines, DphPC and GDPE headgroupsNC3,PO4,OH as blue,orange and red balls, tethers as violet sticks, and spacersas tan sticks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.6 Numerically predicted pore conductance Gp, defined in (4.8). (a) provides thepredicted pore conductance Gp computed using the GPNP, PNP, and EM modelspresented in Sec. 4.3.2. (b) is the predicted Gp for different tethering reservoirdiffusivities. The geometry of the pore is given in Fig. 4.3 with the parametersof the governing equations and boundary conditions provided in Table 4.2. . . . . 754.7 Numerically predicted electrical energy Wes, (4.9), required to form an aqueouspore. (a) compares the predicted Wes computed using the GPNP, PNP, EM, andLM models defined in Sec. 4.3.2 for the transmembrane potential of Vm = 500mV. (b) presents estimates of Wes computed using the GPNP for the transmem-brane potentials listed. (c) provides estimates of Wes computed using the GPNPfor Vm = 500 mV for different tether reservoir diffusivities. The parameters ofthe governing equations and boundary conditions can be found in Table 4.2. . . . 764.8 Numerically predicted Gp and Wes computed using the GPNP model in Sec. 4.3.2with parameters defined in Table 4.1 and Table 4.2. The experimentally mea-sured conductance (BLMs) is obtained from [5–7]. . . . . . . . . . . . . . . . . . 774.9 The measured and predicted impedance of the spacer surface. All predictionsare computed using (4.16) with the parameters defined in Table 4.3. . . . . . . . 784.10 The measured and predicted current response of the spacer surface. All predic-tions are computed using (4.15) with the parameters defined in Table 4.3. . . . . 794.11 The measured and predicted impedance of the 10% tether density DphPC bi-layer membrane. All predictions are computed using (4.17) with the parametersdefined in Table 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.12 The measured and predicted impedance of the 1% tether density DphPC bi-layer membrane. All predictions are computed using (4.17) with the parametersdefined in Table 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80xiiiList of Figures4.13 The measured and predicted impedance of the 1% tether density S. cerevisiaeand E. coli membranes. All predictions are computed using (4.17) with theparameters defined in Table 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.14 The measured and predicted impedance of the 10% tether density S. cerevisiaeand E. coli membranes. All predictions are computed using (4.17) with theparameters defined in Table 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.15 Panels (a) to (d) provide the experimentally measured and numerically predictedcurrent response I(t) for 1%, 10%, and 100% tethered DphPC membranes. In(a) the drive potential Vs(t) is defined by a 1 ms linearly increasing potentialwith a slope of 10 V/s, 40 V/s, 70 V/s, and 90 V/s followed by a 1 ms linearlydecreasing potential with identical slope. In (b) Vs(t) is a 5 ms linearly increasingpotential with slope of 50 V/s, 200 V/s, 300 V/s, and 450 V/s followed by a 5ms linearly decreasing potential with identical slope. In (c) Vs(t) is defined bya 8 ms linearly increasing potential with a slope of 100 V/s followed by a 8 msdecreasing potential with identical slope. In (d) Vs(t) is a step of 50 mV, 100mV, 150 mV, and 200 mV. All predictions are computed using equation (4.14)with the parameters defined in Table 4.3. . . . . . . . . . . . . . . . . . . . . . . 824.16 The measured and predicted current, voltage potentials, membrane resistance,and pore radii for the drive potential Vs(t), defined at the beginning of thissection, for the 10% tether density DphPC bilayer membrane. (a) is the measuredand predicted current, (b) the predicted transmembrane Vm and double-layerpotential Vdl defined in (4.14), (c) is the estimated membrane resistance, and (d)the estimated maximum rmax radius, and mean pore radius r. All predictionsare computed using (4.14) and (4.12) with the parameters defined in Table 4.3. . 834.17 Experimentally measured and numerically predicted current I(t) (a), and mem-brane resistance Rm = 1/Gm (b) for the drive potential Vs(t) is defined by alinearly increasing potential of 100 V/s for 5 ms proceeded by a linearly decreas-ing potential of -100 V/s for 5ms. The tethering densities 1% and 10% correspondto the DphPC bilayer and the 100% corresponds to the DphPC monolayer. Allpredictions are computed using (4.14) and (4.12) with the parameters defined inTable 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85xivList of Figures4.18 Experimentally measured and numerically predicted current response I(t) for the10% tethering density DphPC membrane, panels (a) and (b), and the 1% tetherdensity DphPC membrane, panels (c) and (d). In panels (a) and (c), the drivepotential Vs(t) is defined by a 1 ms linearly increasing with a rise time of 50 to500 V/s in steps of 50 V/s proceeded by a linearly decreasing potential of -50 to-500 V/s in steps of -50 V/s for 1 ms. In panels (b) and (d), the drive potentialVs(t) is defined by a 5 ms linearly increasing potential for 10 to 100 V/s in stepsof 10 V/s proceeded by a linearly decreasing potential of -10 to -100 V/s in stepsof -10 V/s for 5 ms. The numerical predictions are computed using (4.14) and(4.12) with the parameters defined in Table 4.3. . . . . . . . . . . . . . . . . . . . 864.19 The measured and predicted current response of the 1% and 10% tether densityDphPC, S. cerevisiae, and E. coli membranes. The excitation potential Vs isdefined by a linear ramp of 100 V/s for 5 ms followed by a -100 V/s for 5ms. Cell 1 and Cell 2 denote the flow cell number of the CED in which themeasurement was made. All predictions are computed using equation (4.14)with the parameters defined in Table 4.3. . . . . . . . . . . . . . . . . . . . . . . 884.20 Normalized particle density computed from CGMD for the 0% DphPC tetheredmembrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.21 Normalized particle density computed from CGMD for the 25% DphPC tetheredmembrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.1 Fractional order macroscopic model of the ERP for measuring embedded ionchannel conductance Gc. Vm is the membrane potential, Vs the applied potential,and I(t) the current response. The circuit parameters are described in Sec. 5.3. . 965.2 Lumped circuit model of the engineered tethered membrane with cells. Vs isthe driving voltage, Vtm is the potential across the cell membrane adjacent tothe bulk electrolyte, Vbm is the the potential across the cell membrane adjacentto the tethered membrane surface, and Vm is the tethered membrane potential.The circuit parameters are described in Sec. 5.3. . . . . . . . . . . . . . . . . . . 985.3 Experimentally measured and numerically predicted electrophysiological responseof NaCHBac ion channels in the ERP. The numerical predictions are computedusing the model in Fig. 5.1 with the parameters defined in Table 4.3. Panel(a) is the drive potential Vs(t). (b) Experimentally measured and numericallypredicted current response. (c) Predicted transmembrane potential Vm(t). (d)Predicted total aqueous pore Gm(t) and NaChBac conductance Gc(t). . . . . . . 99xvList of Figures5.4 Panel (a) presents the experimentally measured and numerically predicted cur-rent response I(t) for a sawtooth pulse drive potential Vs(t) with a slope of 250V/s for 2 ms. (b) is the computed membrane conductance Gm, Gbm the cellsmembrane conductance adjacent to the tethered membrane, and Gtm the con-ductance of the cell membrane facing the bulk electrolyte solution computedusing the current response in (a). (c) presents the experimentally measured andnumerically predicted current response I(t) for a sawtooth pulse drive poten-tial Vs(t) with a slope of -250 V/s for 2 ms. (d) is the computed membraneconductance Gm, Gbm the cells membrane conductance adjacent to the tetheredmembrane, and Gtm the conductance of the cell membrane facing the bulk elec-trolyte solution computed using the current response in (c). All predictions arecomputed using (4.14) and (4.12) with the parameters defined in Table 4.3. . . . 101xviAcknowledgementsFirst and foremost, I would like to express my sincere gratitude to my academic advisor, Prof.Vikram Krishnamurthy, for his exceptional guidance, constant inspiration and support, andinsightful suggestions that made this work a success. His enthusiasm for tackling new ideas wasalways contagious and a source of motivation.I am also grateful to Dr. Bruce Cornell for his time, effort, and for numerous helpful discus-sions. His comments, suggestions and attention to details have resulted in much improvement,and were crucial for the completion of this work.A sincere thank you to the University of British Columbia for providing the tools andscholarships necessary to support me during my studies.Finally, I am forever grateful for the cheerful company of my friends and colleagues inthe Statistical Signal Processing Laboratory at UBC who I wish not to list for fear that Iwill inadvertently omit someone. Last, and certainly not least, I thank my family for theireverlasting love, enthusiasm for science, and encouragement to pursue my dreams.William HoilesUBC, VancouverFebruary 2015xviiDedicationDedicated to my family and friends for making me who I am.xviii1Introduction1.1 OverviewAll cells contain semi-permeable biological membranes which have a myriad of known func-tions [1]. The membrane provides an interface between the cell and its environment as il-lustrated in Fig. 1.1. Biological membranes are dynamic and respond to their environmentas a result of an external stimulus. The membrane is composed of three primary components:lipids, macromolecules, and the cytoskeletal filaments. The membrane contains numerous typesof lipids, dispersed heterogeneously throughout, which effect the permeabilization, organizationof macromolecules on the membrane surface, cell shape, and organelle function [1]. Macro-molecules perform a variety of function in the membrane [1]. Given the low permeability ofbiological membranes, a primary task of macromolecules is to transport molecules from theextracellular fluid to the cytosol in the interior of the cell. Additionally there exist othermacromolecules which act as molecular receptors, detect vesicles carrying cellular cargo, andfacilitate the transport of other macromolecules across the membrane to the cytosol [1]. Thecytoskeletal filaments provide structural support for the membrane and control the shape ofcells. Given the highly complex dynamics involved in natural cell membranes, there is a needfor artificial cellular membranes which facilitate the study of specific molecular events in con-trolled environments. Additionally, for sensing applications, such artificial cell membranes canbe engineered with embedded macromolecules to detect specific analyte molecules of interest.To perform measurements using artificial cell membranes requires a bioelectronic interfacewhich connects the biological system with electrical instrumentation. In biological systemssignal communication is performed using charge carriers (i.e. ions) rather than electrons. Anumber of resulting differences differences include:• the diffuse three dimensional distribution of ions in solution relative to the delocalizedproperty of electrons in metals;• the slow kinetics of ions in solution compared to the relatively instantaneous kinetics ofelectrons in metals.These effects have been recruited by nature to be the controlling phenomenon in the time11.1. OverviewExtracellular FluidCytoskeletal filamentsCytosolLipids MacromoleculeFigure 1.1: Schematic of a natural biological membrane. The extracellular fluid represents thecontent outside the cell, and the cytosol is the interior of the cell with the membrane separatingthe two domains. Note that biological membranes are composed of thousands of differentcomponents (macromolecules, lipids, chemical species)–refer to [1] for a detailed description.dependent currents in cellular action potentials, in the propagation of action potentials alongnerve fibers, and in the transduction mechanisms of the many sensory functions found in sentientlife. The bioelectronic interface is the physical barrier that must be understood and controlledto permit communication between the complex ephemeral systems of biology and the far faster,better defined, and more robust inorganic world of electronics. An increasingly powerful toolin achieving this understanding is the tethered membrane which incorporates components ofreal cell membranes. The tethered membrane is designed to mimic the physiological responseof real cell membranes.In this thesis the novel engineering and predictive models of four related measurement plat-forms, namely the Ion Channel Switch (ICS) biosensor, Pore Formation Measurement Platform(PFMP), the Controlled Electroporation Measurement Device (CED) , and the Electrophys-iological Response Platform (ERP) are presented. The four measurement platforms employan engineered tethered membrane that mimics the electrophysiological properties of real cellmembranes, and a gold electrode bioelectronic interface to which electrical instrumentation isconnected. Common to all four platforms is that the measurements are performed by estimat-ing the time-dependent conductance of the engineered tethered membrane which is dependenton the bioelectronic interface, and the ensemble of aqueous pores and conducting ion-channelspresent.The use of inert gold electrodes as the bioelectronic interface is superior to redox activeelectrodes for two reasons. First, if redox active electrodes are used, the metal will ablate causingthe tethers to dissociate from the electrode surface destroying the membrane [4]. Second, redoxactive electrodes release metal ions into solution which can interfere with the electrophysiologicalresponse of proteins and peptides. The inert gold electrode capacitively couples the electronic21.1. Overviewdomain to the physiological domain without the issues associated with redox electrodes, howeverthe diffusion-limited effects at the electrode surface must be accounted for when modeling thefour tethered membrane measurement platforms.A schematic of the engineered tethered membrane is given in Fig. 1.2. Key features of theengineered tethered membrane are that the experimentalist can select the density of tethers andmembrane composition with the constructed membrane having a lifetime of several months [4,8–12]. The engineered tethered membrane is composed of a self-assembled monolayer of mobilelipids, and a self-assembled monolayer of tethered and mobile lipids. The tethered lipids areanchored to the gold electrode via a benzyl disulphide component which is connected to apolyethylene glycol chain. Spacer molecules are used to ensure the tethers are spread over thegold electrode. The intrinsic spacing between tethers and spacers is maintained by the benzyldisulphide moieties. A time-dependent voltage potential is applied between the electrodes toinduce a transmembrane potential of electrophysiological interest; this results in a current I(t)that is dependent on the charging of the electrical double-layers and the conductance of theengineered tethered membrane.Bulk ElectrolyteGold Electrode- -+ + ++- ++- -+++++-+ - ++-ElectronicsAqueous PoreTethered MembraneCounter Electrode 𝐺𝑝𝐼(𝑡)++ -gAgAAB B+gA MSLOHether-DLPDLP tether-gASPDphPCGDPE MSLBGoFigure 1.2: Overview of the engineered tethered membrane and molecular components. The“Electronics” block represents the electronic instrumentation that produces the drive poten-tial between the gold electrode and gold counter electrode, and records the current responseI(t). Go is a transient aqueous pore. The conducting gramicidin (gA) dimer is shown and iscomposed of two gA monomers. A represents the analyte species, and B the analyte receptor.MSLOH denotes synthetic archaebacterial membrane-spanning lipids, DLP half-membrane-spanning tethered lipids, DphPC and GDPE mobile half-membrane-spanning lipids, MSLBmembrane spanning lipid, and SP a spacer.Unifying ThemeThe unifying theme that underpins all four measurement platforms is that (a) they all use aninert gold bioelectronic interface, (b) they are constructed using a self-assembled engineered31.1. Overviewtethered membrane, (c) their dynamics can be modeled using a fractional order macroscopicmodel coupled with continuum models and coarse-grained molecular dynamics, (d) each sens-ing mechanism relies on measurement of changes in the tethered membrane conductance, and(e) the “validation metric” of the dynamic models are to get parameter values consistent withknown experimental results. A schematic of the four measurement platforms in this thesis areprovided in Fig. 1.3. Measurement platform 1 (ICS) detects changes in membrane conductanceas a function of the switching dynamics of gA dimers, measurement platform 2 (PFMP) de-tects changes in membrane conductance from pore forming proteins and peptides, measurementplatform 3 (CED) detects changes in membrane conductance resulting from an elevated trans-membrane potential promoting the formation of aqueous pores, and measurement platform 4(ERP) detects the electrophysiological response of ion channels and cells.Impedance MeasurementsCurrent ResponsegA gAgAIon-Channel Switch BiosensorAntibody receptorPore Formation Measurement PlatformPeptideControllable Electroporation DeviceAqueous poreElectrophysiological Response PlatformCellFigure 1.3: A schematic diagram of the ICS, PFMP, CED, and ERP tethered membrane devices.The tethered membrane is depicted in gray, the gold interface by the crosshatch pattern. Theunifying theme of all the devices is the use of an inert gold bioelectronic interface and anengineered tethered membrane. Sensing using the ICS and PFMP is performed by measuringthe time-dependent impedance as a result of changes in the concentration of conducting pores.The sensing mechanism of the CED and ERP is performed by measuring the current responseof the devices to a time-dependent excitation potential. The current response is dependent onthe concentration of conducting pores, the polarization dynamics of the membrane surfaces,and the charging dynamics at the surface of the bioelectronic interface.41.2. Main Contributions1.2 Main ContributionsThis section briefly describes the major contributions of the chapters comprising this thesis.The focus is on modeling the dynamics of four tethered membrane platforms for biosensing, thestudy of pore formation, electroporation, and electrophysiological measurement of ion channelsand cells. The novel modeling approaches and key findings are reviewed below for each chapterin the order that they appear in the thesis. A detailed description of the contributions of eachchapter is provided in the individual chapters.1.2.1 Ion Channel-Switch BiosensorChapter 2 focuses on the predictive models of an ion-channel switch (ICS) biosensor developedby Cornell et. al. [10]. The biosensor can detect femto-molar concentrations of target speciesincluding proteins, hormones, polypeptides, microorganisms, oligonucleotides, DNA segments,and polymers in cluttered electrolyte environments [13–16]. This remarkable detection ability isachieved using engineered receptor sites connected to mobile gA monomers (e.g. the gramicidin(gA) monomers freely diffuse on the membrnae surface) and biotinylated lipids in the tetheredmembrane, refer to Fig. 3.1. To use the experimental measurements from the ICS to estimatethe concentration of analyte species of interest requires a dynamic model.The main contributions of Chapter 2 are summarized below:1. A fractional order macroscopic model of the bioelectronic interface of the ICS. The bioelec-tronic interface of the ICS may contain diffusion-limited charge transfer, reaction limitedcharge transfer, and ionic adsorption dynamics. These double-layer charging effects can bemodeled using fractional order operators [17].2. A continuum model composed of the Poisson-Nernst-Planck (PNP) and reaction-diffusionequations for the transport of analyte molecules in the bulk region and on the membranesurface of the ICS.3. Experimental measurements from the ICS and the dynamic model are used to estimatethe concentration of streptavidin, thyroid stimulating hormone (TSH), ferritin, and humanchorionic gonadotropin (hCG).These results set the stage for Chapter 3 for the design and modeling of a pore formationmeasurement platform (PFMP).1.2.2 Pore Formation Measurement PlatformChapter 3 focuses on how tethered membranes can be used to study the pore formation dynam-ics of peptides and proteins for the development of novel drugs, gene delivery therapies, and51.2. Main Contributionscontrolling pore formation in cell-like bioreactors. Pore forming protein toxins are the most po-tent biological weapons in nature and are key to the attack mechanism of Methicillin-resistantStaphylococcus aureus (MRSA), commonly known as the “super bug” bacteria. Rapid detectionof pore-forming protein toxins is vital for effective treatment. Protein toxins diffuse in solutionuntil the protein binds to a specific membrane. The pore forming toxin may oligomerize prior toforming conducting pores in the membrane causing cell lysis and ultimately killing the cell. Asthe toxin produces conducting pores, measurement of the conductance of the engineered mem-brane allows the rapid pathogenic detection of such toxins and can be used to isolate whichmembranes are most susceptible to attack. In Chapter 3 we present experimental measurementsfrom the pore formation measurement platform (PFMP) and a dynamic model composed ofthree-levels of abstraction: molecular, continuum, and macroscopic reaction rate. Using exper-imental measurements from the PFMP and dynamic model the pore formation dynamics ofpeptides and proteins can be estimated.The main contributions of Chapter 3 are summarized below:1. An atomistic-to-observable model of the PFMP for the study of the dynamics of the an-timicrobial peptide PGLa. The model consists of coarse-grained molecular dynamics, ageneralized reaction-diffusion continuum model, and a fractional order macroscopic model.The CGMD model is used to compute the diffusion coefficient used in the continuum model,and gain key insights into the binding of PGLa to the membrane surface, the translocation ofPGLa from the surface bound state to the transmembrane state, and the oligomerization oftransmembrane bound PGLa. The generalized reaction-diffusion continuum model accountsfor the steric effects of large molecules using a “Langmuir” like activity coefficient [18] andincludes a “Langmuir-Hinshelwood” like equation which is classically used to describe surfacebinding and surface reactions of molecules.2. Using experimental measurements from the PFMP and atomistic-to-observable model, weestimate the pore forming reaction pathway of PGLa in charged and uncharged membranesurfaces that mimic prokaryotic, eukaryotic, and archaebacterial membranes. The resultsshow the potency of PGLa for killing negatively charged membranes that are typically foundin prokaryotes. For the first time we show that PGLa not only increases the number ofpores in negatively charged membranes, but the lifetime of conducting pores also increasescompared to the lifetime of PGLa pores in uncharged membranes.In Chapter 2 and Chapter 3 experimental measurements are performed using impedancemeasurements from the tethered membrane. In Chapter 4 we study how applying a time-dependent excitation potential results in the formation of conducting pores in the tetheredmembrane–this phenomenon is known as electroporation.61.2. Main Contributions1.2.3 Controlled Electroporation Measurement DeviceChapter 4 focuses on how tethered membranes can be used to study the process of electro-poration. Electroporation is the phenomenon where aqueous pores form spontaneously in acell membrane when a high transmembrane potential is applied. Electroporation facilitates thepassage of otherwise impermeable molecules across the membrane or into a cell and is usedto catalyse the uptake of chemotherapeutic agents, DNA molecules, and neuron specific pro-teins in drug delivery applications. It is reported in [19] that electroporation can circumventpoor delivery of medications to the central nervous system for Alzheimers, Parkinson’s disease,and brain cancer. Though in wide use, the electroporation process is still poorly understoodhindering the development of novel electrochemotherapy protocols. The CED considered inChapter 4 provides a controllable and physiologically relevant environment for the study ofelectroporation.The main contributions of Chapter 4 are summarized below:1. In contrast to previous molecular dynamics results [20], no anomalous diffusion was detectedat the surface of the membrane or in the tethering reservoir between the gold and membranesurfaces. This allows the transport of ions away from the electrode surface in the CED tobe modeled using continuum theories that do not include fractional order operators [17].For example the Generalized Poisson-Nernst-Planck GPNP model can be used to model theelectrodiffusive dynamics of ions in the tethering reservoir.2. Membrane thickness, lipid diffusion adjacent to the tethering reservoir, pore density, thefree energy barrier for lipid flip-flop, and effective dielectric permittivity of the membraneare dependent on the tether density. This suggests that when performing electroporationtherapies the tethering density (i.e. density of cytoskeletal supports) is an essential criterion.3. The conductance of aqueous pores in the CED is linearly proportional to the radius of thepore and not the radius squared as typically assumed [21]. This unique feature is a result ofthe proximity of the membrane to the gold surface.4. Experimental measurements show that a combination of diffusion-limited charge transfer,reaction limited charge transfer, and ionic adsorption is present at the surface of the goldelectrodes. These double-layer charging effects can be modeled using fractional order oper-ators [17].5. The predictions that we obtain using the atomistic-to-observable model accurately matchactual experimental data for CEDs from DphPC, E. coli, and S. cerevisae lipids for differentexcitation potentials and tethering densities. This supports the use of molecular dynamicsto go from structure to function.These results set the stage for Chapter 5 for the design and modeling of an electrophysio-logical response platform.71.3. Related Work1.2.4 Electrophysiological Response PlatformChapter 5 focuses on how tethered membranes can be used to measure the electrophysiologicalresponse of ion channels and cells. The electrophysiological response platform (ERP) is com-posed of a tethered membrane composed of embedded ion channels or cells which are grown onthe surface of the membrane. An electrical stimulus is applied to the ERP and the response ofthe ion channels and cells is measured. Chapter 5 illustrates how experimental measurementsfrom the ERP and dynamic model can be used to estimate the electrophysiological responseof ion channels and cells to an external stimulus. The electrophysiological measurement of ionchannels can be used to validate ion channel gating models–of importance for the design ofnovel drugs. The electrophysiological measurement of cells allows for the diagnosis of channelo-pathic diseases such as cystic fibrosis and Bartter’s syndrome. Consider that in cystic fibrosis,the cystic fibrosis transmembrane conducting regulator protein blocks the flow of chloride andthiocyanate ions, which cause a decrease in the cell membrane conductance. The ERP utilizesan engineered tethered membrane for the non-invasive electrophysiological measurement of cellswhich can be used to detect such a change in conductance. To ensure sufficient cell coverageof the membrane, required for accurate measurement, the tether lipid composition, density,and number of spacers are adjusted to promote cell growth. It was determined that a 100%tethered archaebacterial based monolayer provides a suitable membrane to promote cell growthfor electrophysiological measurement.The main contribution of Chapter 5 are summarized below:1. Provide a fractional order macroscopic model to study the electrophysiological response ofion channels and cells to an external voltage excitation. The dynamic model accounts forthe effects of electroporation, diffusion limited processes at the electrode surface of the ERP,and the asymmetry of the cell geometry.2. Use experimental measurements from the ERP and dynamic model to study the electrophys-iological response of the voltage-gated NaChBac ion channel. The NaChBac channel wasfirst reported in 2001 [22], and is likely an evolutionary ancestor of the larger four domainsodium channels in eukaryotes [23, 24].3. Use experimental measurements from the ERP and dynamic model to study the electrophys-iological response of skeletal myoblasts. These cells are attractive donor cells for cardiomy-oplasty used to regenerate damaged myocardium tissue produced from acute myocardialinfarction [25, 26].1.3 Related WorkThis section provides a literature review of progress and advances in the fields and subjectsconnected to the main chapters in this thesis.81.3. Related Work1.3.1 Biosensors and Electrophysiological MeasurementIn this section background and related literature associated with the ICS, PFMP, CED, andERP tethered membrane devices are provided.Electrochemical sensors for biological detection typically employ a biological receptor whichchemically binds to the analyte species of interest. Recently integrated complementary metal-oxide-silicon (CMOS) integrated circuits have emerged for the detection of molecular species [27].These sensors are composed of a sensing surface with a gate oxide or coating such as Poly-L-Lysine [28–30]. Key advantages of CMOS based detection devices compared with opticaltechniques is the low-cost, sensitivity, and selectivity [28–30]. In [27] a 3-glycidoxypropyl-trimethoxysilane self-assembled monolayer coating was used to construct a CMOS sensor forthe detection of biotinylated bovine serum albumen with streptavidin. In [28] the CMOS sensorsurface is coated with Poly-L-Lysine and is used to detect for proteins associated with tumorcells. The principal operation of these CMOS devices is to detect for changes in the sensor sur-face capacitance as proteins bind to the surface. This is in contrast to the principal operationof the four tethered membrane platforms presented in this thesis in which the capacitances areconstant, and the principal operation is based on measuring the changes in tethered membraneconductance.1. Ion Channel Switch Biosensor: The Ion Channel Switch (ICS) biosensor discussedin Chapter 2 was originally developed by Dr. Cornell and its construction is detailed in theNature publication [10]. The ICS biosensor is a fully functioning nanomachine constructed outof a tethered artificial cell membrane with moving parts comprising gramicidin (gA) monomersand conducting gA dimer channels. Gold electrodes constitute the bioelectronic interface be-tween the electrical instrumentation and the electrolyte solution. The ICS biosensor can detectfemto-molar concentrations of target species including proteins, hormones, polypeptides, mi-croorganisms, oligonucleotides, DNA segments, and polymers in cluttered electrolyte environ-ments [13–15], and has a lifetime of several months [4, 8–10, 12]. The ICS biosensor has alsobeen used in clinical trials for the detection of Influenza A [14].Previous models of the ICS have focused on using reaction rate theory [31], and a combi-nation of reaction rate theory and the Nernst-Planck equations for advection-diffusion [13, 15].The dynamic models presented in Chapter 2 have three key advantages to those proposedpreviously in [13, 15]:1. the use of surface reaction-diffusion equations to account for the diffusion kinetics ofspecies on the membrane surface of the ICS,2. the use of a fractional order macroscopic model which accounts for the diffusion limitedprocesses present at the electrode surface of the ICS,3. accounting for the equilibrium aqueous pore conductance Go.91.3. Related WorkIn Chapter 2 the proposed model of the ICS is validated using the experimentally measuredmembrane conductance for four important analyte species: streptavidin, thyroid stimulatinghormone (TSH), ferritin, and human chorionic gonadotropin (hCG). The antibody receptorused to bind streptavidin to the membrane surface is biotin. The streptavidin to antibodyreceptor biotin provides an excellent case study for validation of the proposed dynamic model.The concentration of TSH present can be used to detect hypo or hyperthyroidism. Ferritinplays a central role in the transport, storage, and release of iron. As the concentration offerritin increases, this can be directly associated with a pathogenic infection or the presence ofcancer [32]. Here we measure the concentration of ferritin in whole blood (i.e. human bloodfrom a standard blood donation). The concentration of hCG is an excellent indicator of thepresence of blastocyst or mammalian embryogenesis (i.e. pregnancy).2. Pore Formation Measurement Platform: The PFMP is used to detect the pres-ence of pore forming proteins and peptides. Since the membrane surface is engineered to mimicprokaryotic, eukaryotic, and archaebacterial membranes, the PFMP can be used to measurethe specificity of attack of protein and peptide toxins. Therefore, the PFMP can be used forrapid point-of-care detection of pore forming toxins and for inexpensive pharmacology screeningof novel antimicrobial peptides. Other techniques, which do not include tethered membranes,to study pore formation dynamics include: lytic experiments, gel electrophoresis, site-directedmutagensis, and cryoelectron microscopy [33–36]. The benefit of using the PFMP compared tothese methods is that the tethering density, electrolyte composition, membrane composition,and applied transmembrane potential can all be controlled by the experimentalist. Examplesof tethered bilayer lipid membranes for the measurement of pore forming toxins include [37]in which the membrane is composed of diphytanyl chains that are coupled via a glycerol tooligoethylene oxide spacers, and [38] that uses different lipid, anchoring, and spacer compo-nents then the PFMP but employs an identical solvent-exchange membrane formation protocol.In [37, 38] lumped circuit models of the tethered membrane are used to infer if the protein orpeptides bind to the surface. To gain insight into the reaction pathway leading to pore forma-tion requires a dynamic model that accounts for the diffusion of the proteins and peptides insolution, and the reaction-diffusion processes present on the membrane surface. In Chapter 3a three level model is provided which accounts for these effects.3. Controlled Electroporation Measurement Device: Experimental platforms tostudy electroporation include synthetic bilayer lipid membranes and in vitro cells [39, 40].However, synthetic bilayer lipid membranes do not provide a good representation of physiolog-ical systems since the effects caused by the cytoskeletal network are not present [41, 42]. Usingcells provides a physiological system for validation [43, 44]; however, it is impossible to fullydefine the physiological environment which affects properties associated with electroporation.This motivates the need for an engineered tethered membrane platform which gives the experi-mentalist control over tethering density, membrane composition, and physiological environment101.3. Related Workunlike the synthetic lipid membrane and cell based platforms.Models of the electroporation process employ the Smoluchowski-Einstein equation derivedfrom statistical mechanics [41, 45, 46] with the pore energy models given in [47–54]. The poreenergy models are constructed by assuming the membrane is a dielectric and elastic contin-uum [47, 55–60]. The Smoluchowski-Einstein equation is numerically prohibitive to solve, andthe pore energy provided in the literature does not include effects caused by asymmetric elec-trolytes, multiple ionic species, and the Stern and diffuse electrical double layers present. Toovercome these limitations, the electroporation model presented in Chapter 4 is constructedusing asymptotic approximations to the Smoluchowski-Einstein equation and the General-ized PNP (GPNP) for modeling the electrodiffusive dynamics. To couple the results of theSmoluchowski-Einstein equation with the experimentally measured current requires an estimateof the aqueous pore conductance. In [21], assuming symmetric electrolytes and electroneutrality,the aqueous pore conductance was estimated via the Poisson-Nernst-Planck (PNP) system [61].In the tethered membrane platform there exists electrical double layers at the surface of themembrane and electrode contacts [62] which can be modeled using the GPNP [63]. The GPNPis equivalent to the Poisson-Nernst-Planck (PNP) system of equations if steric effects are ne-glected. Recent models for computing pore conductance Gp in the literature use the (PNP),assuming symmetric electrolytes and electroneutrality [21]; and models for computing the elec-trical energy required to form a pore Wes assuming symmetric electrolytes, electroneutrality,and negligible concentration gradients [51]. The evaluation of Gp and Wes using the GPNP ac-counts for the dynamics of electrolytes and the Stern and diffuse layers present in the CED. Notethat near a pore entrance significant nonlinear potential gradients are present which restrictthe current flowing through the pore, this effect is denoted as the “spreading conductance” andis dominant for pore radii significantly larger then the membrane thickness causing the poreconductance to scale proportionally to the pore radius [21, 42, 45, 64, 64–66]. As illustratedin Chapter 4, the pore conductance scales proportionally to the pore radius for all pore radiisuggesting that the “spreading conductance” is dominant as a result of the electrode in prox-imity to the membrane surface and the nonlinear potential gradients present. This effect onlybecomes pronounced for large pores in free floating membranes [21].4. Electrophysiological Response Platform: The ERP is a tethered membrane plat-form designed to measure the electrophysiological response of ion channels and cells.Classical methods for the electrophysiological measurement of ion channels include patch-clamp electrophysiology in mammalian cells and two-electrode voltage clamp in Xenopus oocytes [67].Both of these methods produce extremely information-rich data which can be used to validateion-channel gating models. However, these approaches are labor intensive and require highlyskilled staff to ensure reproducible results are obtained. Embedding ion channels in a con-trollable tethered membrane environment allows the electrophysiological measurement of ionchannels in a platform that requires approximately 20 minutes to form using standard labora-111.3. Related Worktory techniques. The membrane and electrolyte environment of the ERP can be controlled withembedded ion channels to study the voltage-gating dynamics in an environment that mimicsthe native environment of the ion channel. The ERP and dynamic model can therefore be usedfor high throughput drug screening and the validation of ion channel gating models.There are two major methods for measuring the electrophysiological response of cells. Thefirst is to use substrate-integrated microelectrode arrays, and the second is to use sharp or patchmicroelectrodes that puncture the cell [68]. A limitation with invasive cell measurement thatemploy sharp and patch microelectrodes is that a limited number of cells can be analyzed for ashort period of time. Substrate-integrated microelectrode arrays provide a non-invasive methodfor measuring the electrophysiological response of cells however a major challenge when usingthese sensors is to ensure sufficient cell adhesion and coverage [68, 69]. An emerging technologyto ensure cell adhesion is to use a metal electrode coated with a polycationic film onto which anadhesion protein, such as Glycocalyx, is used to bind with the cell membrane [68]. In Chapter 5it is illustrated that using a suitably designed tethered membrane, cells can reach a sufficientcoverage and adhesion to allow for their electrophysiological measurement. Remarkably thisallows the response to be measured using a non-invasive technique in the proximity of a syntheticmembrane that mimics the electrophysiological response of biological membranes.1.3.2 Levels of Abstraction in Tethered Membrane ModelingThis section outlines the levels of abstraction (atomistic to macroscopic) that have been usedto model membrane dynamics and provides insight into the construction of the atomistic-to-observable models presented in this thesis.Key to the development of novel biosensing platforms is an accurate dynamic model ofthe membrane, macromolecules, and the bioelectronic interface. Such a model must link theatomistic dynamics of water, ions, membrane lipids, proteins, and peptides to experimentalmeasurements at the macroscopic time scale. It is essential that dynamical model be based onphysical principles and result in computationally tractable simulation algorithms. The com-putational tools of physics employed in this endeavor, from fundamental to phenomenological,are ab initio molecular dynamics, classical molecular dynamics, coarse-grained molecular dy-namics, continuum theories, and reaction rate theory. These approaches make various levels ofabstractions in replacing the complex reality with a model. Each of these approaches has itsstrengths and limitations and involves a degree of approximation. Fig. 1.4 provides an overviewof the length scale and simulation time horizon achievable by general atomistic to macroscopicmodels.121.3. Related WorkLevels of AbstractionAb-initioMolecular Dynamicsnm fsClassicalMolecular Dynamicsnm nsCoarse-GrainedMolecular Dynamicsnm µsContinuumTheoryµm µsMacroscopicTheorym sFigure 1.4: Schematic diagram illustrating the length and timescale achievable by the atomisticto macroscopic simulation methods in Sec. 1.3.2.1.3.3 Ab-initio Molecular DynamicsAt the lowest level of abstraction is ab-initio molecular dynamics, which combines Newton’slaws and Schro¨dinger’s equations, to model the dynamics of water, ions, membrane lipids,proteins, and peptides. There are three terms in the associated Hamiltonian: the Coulomb in-teractions, electronic part, and the nuclear part [70, 71]. In ab-initio molecular dynamics thereare no free parameters and therefore represents the ultimate tool for modeling tethered mem-brane platforms. However, performing ab-initio molecular dynamics simulations is a formidabletask. Currently, the attainable length and simulation time horizons are on the order of a fewnanometeres and femtoseconds [72].1.3.4 Molecular DynamicsA simplification to ab-initio molecular dynamics is to replace the potential energy function inthe many-body quantum mechanical Schro¨dinger’s equation by a phenomenological one. Atthis level of abstraction the force-field is empirically parameterized to describe the pair-wiseinteraction between ions with the dynamics of ions evaluated using Newton’s equation of mo-tion. This simulation method is known as molecular dynamics [73]. Using molecular dynamicsallows equilibrium thermodynamic and dynamical properties of a system at finite temperatureto be computed where the interatomic forces are not effected by accurate electronic structurecalculations (e.g. chemical bonding processes). Though molecular dynamics only requires theevaluation of Newton’s laws, the typical system size and simulation time achievable is on theorder of nanometers and nanoseconds [74]. For computing important biological parameters intethered membranes the simulation time horizon achievable must be on the order of microsec-onds with a system size of tens of nanometers.1.3.5 Coarse-Grained Molecular DynamicsA simplification to molecular dynamics is to group atoms together into coarse-grained beads,with the bead-to-bead interactions empirically parameterized allowing their dynamics to beevaluated using Newton’s equation of motion. This method of abstraction is known as coarse-grained molecular dynamics (CGMD). In CGMD, the force-field is empirically parameterized by131.3. Related Workmatching the bead dynamics to appropriate experimental data and all-atom molecular dynamicssimulations. This allows the CGMD model to achieve simulation time horizons of microsecondswith a system size of tens of nanometers.In this thesis the MARTINI force-field [75, 76] is utilized for all CGMD simulations of thetethered membranes. The MARTINI model is based on taking four heavy atoms (e.g. fourcarbon atoms and hydrogen, or four water molecules) and representing this structure by asingle coarse-grained bead. The MARTINI force-field contains 18 possible beads which canbe used to represent lipids and amino-acid sequences [74]. The MARTINI force-field has beenused for the study of tethered membranes [77, 78], and the oligomerization of peptides (referto citations in [74]). In [78] the membrane consists of DOPC lipids, and pegylated DOPClipids with different tethering lengths. The simulation results show that decreasing the lengthof tethers increases the stability of the membrane. In [77] the membrane consists of DPPClipids, pegylated DPPC lipids, and surface with hydroxyl-terminated β-mercaptoethanol whichis typically used to prepare the gold surface for tethered membrane assembly. The results in[77] provide the formation dynamics of the tethered membrane for different lipid concentrationsand show that once a sufficient concentration of lipids is present, then a tethered bilayer lipidmembrane will self-assemble.The CGMD model of the PFMP and CED presnted in this thesis are composed of cus-tom coarse-grained structures which model the bioelectronic interface, tethers, spacers, lipids,and antimicrobial peptide PGLa. These structures are based on those reported in [74, 77, 78]and constructed using the methods provided in [2, 75]. The CGMD models of the tetheredmembranes are constructed with a focus on computing important biological parameters (e.g.diffusion, line tension, surface tension) which can be used in continuum and macroscopic mod-els. This allows the results from the CGMD simulations to be validated using experimentalmeasurements from the PFMP and CED.1.3.6 Continuum TheoriesThe next level of abstraction from CGMD is to apply the mean-field approximation [79] whichallows the dynamics of ions to be modeled from a continuum theory–ions are treated not asdiscrete entities but as continuous charge densities that represent the space-time average of themicroscopic motion of ions. This allows continuum models to achieve simulation time horizons ofmicroseconds with a system size of micrometers. The most well-known continuum theory modelfor ion transport is certainly the Poisson-Nernst-Planck system of equations which combinesthe Poisson equation from electrostatics, and the Nernst-Planck equation for diffusion [80–90]. Primarily in a biological context, the PNP theory is used to model ion transport throughion channels and nanopores [81–85, 91–93]. In Chapter 2 and Chapter 3 continuum theoriesare employed to relate the diffusion and chemical reactions of molecular species to changes in141.4. Thesis Outlinemembrane conductance. In Chapter 4 the GPNP continuum model is employed to computethe conductance of aqueous pores, and electrical energy required to form a pore in the tetheredmembranes. These are necessary for computing the dynamic response of the tethered membraneto an elevated transmembrane potential.1.3.7 Reaction Rate TheoryAt the highest level of abstraction is macroscopic theory. In this approach a phenomenologicalmodel is developed to model the dynamics where the model parameters do not have to have adirect physical interpretation. Given the parameters of the macroscopic model do not relate todirect physical interpretation, the length scale and time scale achievable are arbitrarily large.In this thesis the fractional order macroscopic model is utilized to estimate important biologicalparameters such as membrane conductance and polarization from experimental measurementsfrom the tethered membrane platforms.Note that the atomistic-to-observable model links the three levels of abstractions fromcoarse-grained molecular dynamics to the macroscopic theory, refer to Fig. 1.4.1.4 Thesis OutlineThis thesis is composed of six chapters as outlined below:• Chapter 2 introduces the formation and operation of the ion-channel switch (ICS) biosen-sor used to measure the concentration of specific analyte molecules. A dynamic modelof the ICS is constructed using macroscopic and continuum models. The fractional ordermacroscopic model accounts for the diffusion-limited charge transfer that is present at thegold bioelectronic interface of the ICS biosensor. The transport of analyte molecules tothe sensing surface is modeled using the Poisson-Nernst-Planck system of equations, withthe surface reactions accounted for using reaction-diffusion equations. This chapter con-cludes by illustrating how experimental measurements from the ICS and dynamic modelcan be used to estimate the concentration of Streptavidin, TSH, Ferritin, and hCG insolution.• Chapter 3 introduces the formation and operation of the pore formation measurementplatform (PFMP) of use for measuring the pore formation dynamics of proteins and pep-tides. An atomistic-to-observable dynamic model composed of coarse-grained moleculardynamics, generalized reaction-diffusion continuum model, and the macroscopic fractionalorder model. The PFMP and atomistic-to-observable model are validated by studyingthe pore formation dynamics of the protein toxin α-Hemolysin [94] from Staphylococcusaureus. This chapter concludes with the numerical and experimental study of the poreformation dynamics of the antimicrobial peptide PGLa.151.4. Thesis Outline• Chapter 4 introduces the controlled electroporation measurement device (CED) , a pre-cisely controllable platform for electroporation studies. An atomistic-to-observable dy-namic model comprising of three levels of abstraction: coarse-grained molecular dynam-ics (CGMD) that incorporates membrane tethers, a Generalized Poisson-Nernst-Planckcontinuum model, and a fractional order macroscopic model is provided. The chapter con-cludes with using the ERP and atomistic-to-observable model to gain key insights intothe effect the bioelectronic interface, archaebacterial, Escherichia coli, and Saccharomycescerevisiae lipids, and electrolyte concentration, have on the process of electroporation.• Chapter 5 introduces the formation and operation of the electrophysiological responseplatform (ERP). The ERP is designed to measure the physiological response of ion chan-nels and cells to an electrical stimulus. A fractional order model is presented to allow theresponse of the ion channels and cells to be estimated from experimental measurementsfrom the ERP. The chapter concludes with numerical and experimental measurementsof the electrophysiological response of the voltage gated NaChBac ion channel, and ofskeletal myoblast cells.Finally, Chapter 6 briefly summarizes the results, provides concluding remarks, and com-ments on the future directions for research in the areas related to the material presented in thisthesis.162Ion Channel Switch Biosensor2.1 IntroductionIn this chapter the construction and dynamic modeling of the Ion Channel Switch (ICS) biosen-sor is presented. A key feature of the ICS biosensor is that the detection is performed by mea-suring the time-dependent conduction of the engineered tethered membrane which is dependenton the ensemble of aqueous pores and conducting gA dimers present. Using specific molecularcomponents and drive potentials, the ICS can be designed to detect specific analyte moleculesof interest.Recent advances in detecting biomolecules include the nanogap biosensor [95, 96] and thenanoneedle biosensor [97]. The nanogap biosensor relies on detecting impedance changes of theelectrode surface which is proportional to the concentration of target molecules. An issue withthe nanogap biosensor and similar sensors is that spurious electrochemical reactions resultingfrom proteins and ions binding to the electrode surface can interfere with measurements. As thenanogap biosensor utilizes a redox active electrode (e.g. Ag/AgCl) [96], the electrode ablatesand releases metal ions into solution which can conformationally change the biomolecules be-ing detected resulting in measurement errors. The nanoneedle biosensor utilizes the change inconduction between two poly-silica phosphorous doped electrodes sandwiched between silicondioxide. The electrodes are placed 30 nm apart allowing for the detection of biomolecules. Assilicon is poorly soluble it does not release harmful ions into solution; however, since silicondioxide has an isoelectronic point of 3 (i.e. the pH at which silicon dioxide carries no net elec-trical charge), the adsorption of certain proteins and peptides on the surface is a possibility [97].Given the electrode size and detection mechanism of the nanoneedle biosensor, it is difficult toperform concentration estimates in cluttered electrolyte environments.To overcome these limitations, the ICS biosensor employs an inert bioelectronic interfaceand an engineered tethered membrane for detection and measurement. The electrical instru-mentation of the ICS is connected to the electrolyte solution via gold electrodes. Using goldelectrodes as the bioelectronic interface provides a superior interface as compared with redoxactive electrodes for two reasons [4]. First, if redox active electrodes are used, the metal willablate causing the tethers to dissociate from the electrode surface destroying the membrane.172.1. IntroductionSecond, redox active electrodes release metal ions into solution which can interfere with theelectrophysiological response of proteins and peptides. The inert gold electrode capacitivelycouples the electronic domain to the physiological domain without the issues associated withredox electrodes, however the capacitive effects of the electrode must be accounted for whenmodeling the ICS biosensor.A schematic of the ICS is given in Fig. 1.2. Useful for experimentalists is the fact the ICS canbe designed with specific binding cites with the membrane having a lifetime of several months [4,8–12]. The engineered tethered membrane is composed of a self-assembled monolayer of mobilelipids and gA monomers, and a self-assembled monolayer mobile lipids and gA monomers. Thetethered components are anchored to the gold electrode via polyethylene glycol chains. Spacermolecules are used to ensure the tethers are evenly spread over the gold electrode. The intrinsicspacing between tethers and spacers is maintained by the benzyl disulphide moieties which bondthe spacers and tethers to the electrode surface. A time-dependent voltage potential is appliedbetween the electrodes to induce a transmembrane potential of electrophysiological interest;this results in a current I(t) related to the charging of the double-layers and the conductanceof the engineered tethered membrane.The ICS biosensor is capable of detecting femto-molar concentrations of target species in-cluding proteins, hormones, polypeptides, microorganisms, oligonucleotides, DNA segments,and polymers in cluttered electrolyte environments [13–15]. This remarkable detection abilityis achieved using engineered receptor sites connected to mobile gA monomers and biotinylatedlipids in the tethered membrane, refer to Fig. 1.2. By measuring the dynamics of the membraneconductance, the concentration of a specific analyte can be estimated. In reference to Fig. 1.2,the mobile gA monomer is tethered to a biological receptor such as a nucleotide or antibodywhich binds to specific target species. In the neighborhood of the mobile gA, a tethered mono-layer lipid is present with the tethered receptor present in contact with the analyte solution.When the receptor binds to the analyte, the mobile gA monomer diffuses to the tethered lipidcausing the conducting gA dimer to break. As an ensemble of gA dimers dissociate, the con-ductance of the membrane decreases. Measurement of the conductance change allows boththe detection of the analyte species and a estimate of the concentration of the analyte speciesin cluttered environments as the receptor is designed to bind to a specific target species. Torelate the analyte concentration to changes in membrane conductance requires the use of anelectrodiffusive model for the analyte coupled with the surface reactions present at the tetheredmembrane surface.2.1.1 Main Results and Chapter OrganizationThe main contribution of this chapter is an electrodiffusive model which can be used to es-timate the concentration of analyte molecules from experimental measurements from the ICS182.2. Ion Channel-Switch Biosensor: Construction and Formationbiosensor. The electrodiffusive model is composed of a fractional order macroscopic model anda continuum reaction-diffusion model. Sec. 2.2 provides the construction and formation of theICS biosensor. Sec. 2.3 the dynamic model for estimating the analyte concentration is pre-sented. In Sec. 2.4 experimental measurements from the ICS and dynamic model are used tomeasure the concentration of four important analyte species: streptavidin, thyroid stimulatinghormone (TSH), ferritin, and human chorionic gonadotropin (hCG). The streptavidin to anti-body receptor biotin provides an excellent case study for validation of the proposed dynamicmodel as the binding affinity for streptavidin to biotin is extraordinarily high [98]. The concen-tration of TSH present can be used to detect hypo or hyperthyroidism. Ferritin plays a centralrole in the transport, storage, and release of iron. As the concentration of ferritin increases, thiscan be directly associated with a pathogenic infection or the presence of cancer [32]. Here wemeasure the concentration of ferritin in whole blood (i.e. human blood from a standard blooddonation). The concentration of hCG is an excellent indicator of the presence of blastocyst ormammalian embryogenesis (i.e. pregnancy). Closing remarks are provided in Sec. 2.5.2.2 Ion Channel-Switch Biosensor: Construction andFormationThe engineered tethered membrane of the ICS is supported by a 25×75×1 mm polycarbonateslide. Six 100 nm thick sputtered gold electrodes, each with dimensions 0.7×3 mm, rest onthe polycarbonate slide. Each electrode is in an isolated flow cell with a common gold returnelectrode. The formation of the tethered membrane on the gold electrode is performed in twostages using a solvent-exchange technique [10, 99, 100].Stage 1: The first stage of formation involves anchoring of the tethers and spacers to the goldsurface. The tethers provide structural integrity to the membrane and mimic the physiologicalresponse of the cytoskeletal supports of real cell membranes. The spacers laterally separatethe tethers allowing patches of mobile lipids to diffuse in the membrane. The tethers andspacers both contain benzyl disulphide components (i.e. MSLOH, DLP, ether-DLP, tether-gA, MSLB, and SP in Fig. 1.2). The benzyl disuphide bonds to the gold surface with thedisulphide bond maintained [10, 14]. This bonding structure has been detected experimentallyfrom X-ray photoelectron spectra. The use of the disulphide has the advantage that the thiolsdo not oxidize on storage allowing the membrane to have a lifetime of several months. Fromexperimental measurements, the electrodesorption of the thiol to gold bond is negligible forelectrode potentials below 800 mV [101].To form the anchoring layer, an ethanolic solution containing 370 µM of engineered ratios ofbenzyl disulphide components is prepared. The ratio of benzyl disulphide components definesthe tethering density of the membrane. For example, for a 10% tethered membrane, for every 9spacer molecules there is 1 tether molecule. This solution is exposed to the gold surface for 30192.2. Ion Channel-Switch Biosensor: Construction and Formationmin, then the surface is flushed with ethanol and air dried for approximately 2 min. Note that inthe special case of 100% tethering, the engineered tethered membrane is composed of a tetheredarchaebacterial based monolayer with no spacer molecules. As experimentally illustrated in [8],it is not possible to construct a 0% tethered membrane as any formed membrane binds to thegold surface. As the electrolyte reservoir separating the membrane and electrode surface isrequired for the normal physiological function of the membrane, and noting that all prokaryoticand eukaryotic cell membranes contain cytoskeletal supports with a 1% to 10% tether density,the inability to construct a 0% tethered membrane is of little importance.Stage 2: The second stage involves the formation of the tethered membrane. A solutioncontaining a mixture of mobile lipids is brought into contact with the gold bonded componentsfrom Stage 1. Several lipid solvents can be used [9, 10], however in most cases the lipids selectedto form the bilayer are soluble in ethanol. As an example, let us consider the formation of a70% DphPC and 30% GDPE mixed tethered membrane. 8 µL of 3 mM of the 70% DphPCand 30% GDPE ethanolic solution is added to the flow chamber. The ethanol solution alsocontains biotinylated gA monomers (tether-gA in Fig. 1.2) with the biotin linked to the gAmonomer via a 5-aminocaproyl linker. The solution is incubated for 2 min at 20oC in which thetethered membrane forms. After the 2 min incubation period, 300 µL of phosphate bufferedsaline is flushed through each flow chamber. The tethered membrane is equilibrated for 30 minprior to performing any experimental measurements. For the detection of streptavidin a biotinreceptor is used. The associated antibody fragments used to detect ferritin, TSH, and hCGare the anti-ferritin Fab′, thyrotropin binding inhibitory immunoglobulin, and immunoglobulinG respectively. Details on how the antibodies are connected to the MSLB and mobile gAmonomers, in Fig. 1.2, is provided in [10].Quality Control and Measurement: The quality of the tethered membrane is measuredcontinuously using an SDx tethered membranes tethaPodTM swept frequency impedance readeroperating at frequencies of 1000, 500, 200,100,40,20,10,5,2,1,0.5,0.1 Hz and an excitation po-tential of 20 mV (SDx Tethered Membranes, Roseville, Sydney). Custom drive potentials areproduced and the resulting current recorded using an eDAQTM ER466 potentiostat (eDAQ,Doig, Denistone East) and a SDx tethered membrane tethaPlateTM adaptor to connect to theassembled electrode and cartridge. The defect density in the formed membrane can be estimatedfrom the impedance measurements using the protocol presented in [102]. For all experimentalmeasurements, the membrane contained negligible defects. To detect the presence of electrodes-orption and release of portions of tethered membrane into solution, the capacitance of the goldelectrode and membrane is monitored. Using the experimentally measured impedance and dy-namic model in Sec. 2.3.1, the integrity of the sulphur-gold bond and membrane is ensured bycomparing the associated capacitances before and after all experimental measurements.202.3. Dynamic Model of the Ion Channel-Switch Biosensor2.3 Dynamic Model of the Ion Channel-Switch BiosensorIn this section a dynamic model is constructed for the ICS biosensor. The dynamic modelof the ICS is composed of a fractional order macroscopic model, and a continuum model forelectrodiffusion. The dynamic model can be used to estimate the concentration of target analytespecies given experimental measurements from the ICS biosensor.2.3.1 Fractional Order Model of the Bioelectronic InterfaceIn this section a fractional order macroscopic model is provided to compute the current responseof the ICS. Fractional order operators are utilized in the model as the gold surface bioelectronicinterface of the ICS may contain diffusion-limited charge transfer, reaction limited charge trans-fer, and ionic adsorption dynamics. These double-layer charging effects can be modeled usingfractional order operators [17].The ICS is composed of three distinct regions: the bioelectronic interface at the gold elec-trodes, the tethered membrane, and the bulk electrolyte solution. The membrane is assumed tobe polarizable and to also contain aqueous pores as a result of random thermal fluctuations–thatis, random thermal fluctuations allow the energy barrier to be crossed for the conformationalchange of lipids allowing the formation of transient aqueous pores. This allows the tetheredmembrane to be modeled by an effective permittivity with capacitance Cm in parallel withthe tethered membrane conductance Gm(t) [3, 8, 16]. Gm is dependent on the population ofaqueous pores and conducting gA dimers present. Since a voltage excitation of 30 mV is used,the population of aqueous pores is constant and is accounted for by the associated equilibriummembrane conductance Go. The total conductance of all gA dimers, denoted GD(t), is propor-tional to the population of gA dimers present (i.e. GD(t) ∝ cds(t) with cds(t) the concentrationof gA dimers). To link the membrane conductance Gm to the target analyte concentrationrequires a model that accounts for the electrodiffusive effects of the analyte in the electrolyte,and the surface reactions present on the tethered membrane surface of the ICS biosensor. Thebulk electrolyte solution is assumed to be purely ohmic with a resistance Re. There existsan electrical double layer [103] at the bioelectronic interface of the ICS which can be modeledusing a capacitor if diffusion-limited charge transfer, reaction limited charge transfer, and ionicadsorption dynamics are not present. If these double-layer charging effects are present thenthe bioelectronic interface can be modeled using a constant-phase-element composed of a ca-pacitance Cdl and the fractional order operator p. If p < 1 then a diffusion-limited process ispresent, and if p = 1 then a diffusion-limited process is not present. The electrode capacitanceadjacent to the tethered membrane is denoted by Cbdl, and the counter electrode capacitanceby Ctdl. An excitation potential Vs(t) is applied across the two electrodes of the ICS and thecurrent response I(t) is measured. The fractional order macroscopic model of the ICS biosensor212.3. Dynamic Model of the Ion Channel-Switch Biosensoris given by (Fig. 2.1):dVmdt = −(1CmRe+ GmCm)Vm −1CmReVdl +1CmReVs,dpVdldtp = −1CdlReVm −1CdlReVdl +1CdlReVs, (2.1)I(t) = 1Re(Vs − Vm − Vdl), (2.2)where Cdl is the total capacitance of Ctdl and Cbdl in series with p in (2.1) denoting the orderof the fractional order operator, Vm is the transmembrane potential, and Vdl is the double-layerpotential. Note that if p < 1 then the SI units of Cdl are s(p+3)/pA2/pm2/pkg1/p. Given the drive potentialVs(t), and the static circuit parameters Ctdl, Cbdl, Cm, and Re, the membrane conductanceGm(t) can be estimated from the measured current I(t).Cbdl+− VsGm Cm+−VmReCtdlI(t)Figure 2.1: Fractional order macroscopic model of the ICS biosensor. The circuit parametersare defined in Sec. 2.3.1.Using a sinusoidal drive potential Vs(t) = Vo sin(2pift) with frequency f and magnitudeVo below 50 mV, the current response of the ICS can be computed using a set of algebraicequations. Converting (2.1) and (2.2) into the complex domain with Vs(t) = Vo sin(2pift), thecurrent response of the ICS is given by:I(f) = Vo[Re +1Gm + j2pifCm+ 1(j2pif)pCdl]−1. (2.3)In (2.3), j denotes the complex number√−1. Note that the impedance Z(f) of the ICS biosen-sor is given by the expression in [·] of (2.3). Assuming Gm(t) is static during the measurementof I(f), the membrane conductance Gm(t) can be computed using a least-squares estimatorwith a cost function given by the difference between the measured current and the computedcurrent from (2.3). Note that the parameters Re, Cm, p, and Cdl are constant over the frequencyrange of measurement (i.e. 0.1 Hz to 1 kHz).222.3. Dynamic Model of the Ion Channel-Switch Biosensor2.3.2 Continuum Model of the Bioelectronic InterfaceThe detection of analyte molecules by the ICS biosensor is performed by detecting changesin the tethered membrane conductance resulting from the molecules of interest binding to theengineered binding sites. As molecules bind the population of conducting gA dimers decreaseswhich decreases the total conductance of the tethered membrane. There are two processespresent. First, the analyte molecule undergoes an electrodiffusive process in the bulk solutionabove the tethered membrane surface. Second, a surface reaction-diffusion process occurs whichgoverns the population of conducting gA dimers in the tethered membrane. The electrodiffusiveprocess and surface reaction-diffusion process are coupled with the analyte molecules bindingto the engineered binding sites.The study of the electro-diffusion of charged particles such as ions, electrons, or colloids inthe presence of an applied external electric field is of great importance in a number of disciplines.The motion of electrons and holes in semiconductors subject to an imposed external electricfield is of importance to the design of modern electronic components such as transistors, diodes,and infrared lasers [91]. In the biological applications, the process of interest is the flow of inor-ganic ions (K+, Na+, Ca2+, Cl−, etc.) through pores in lipid bilayer membranes. Biochemicalsignal communication in living organisms heavily relies on the transport and concentration ofinorganic species.The most well-known continuum theory model for ion transport is certainly the Poisson-Nernst-Planck system of equations which combines the Poisson equation from electrostatics,and the Nernst-Planck equation for diffusion [80–90]. Primarily in a biological context, thePNP theory is used to model ion transport through ion channels and nanopores [81–85, 91–93]. Mathematical models for a biological system that considers the transport of ions, chargedmolecules, and molecular charged species, and a microelectronic system which considers thetransport of electrons and holes can be linked using the PNP theory. In [89], this idea of bio-electronic interfacing was studied using the steady-state PNP theory in which an electrogeniccell was placed on the top of a field-effect-transistor gate. Interested readers are referred to[104] for an introduction to bioelectronic interfacing using semiconductor devices.Here we provide the intuition behind the PNP theory and what conditions must be satisfiedfor its application. Two common methods for the derivation of the PNP theory are to beginfrom either the electrochemical potential of equilibrium thermodynamics or using the proper-ties of diffusion and electrostatics [105]. Here we provide the derivation using the properties ofdiffusion and electrostatics. The derivation begins by assuming the ion interactions and contin-uum descriptions of concentration and electric potential are valid–the mean-field approximationholds [106]. Consider that the transport behavior of ions is driven primarily by a diffusive flux232.3. Dynamic Model of the Ion Channel-Switch BiosensorJ id, and an electrical-migration flux J ie where i denotes the ionic species, and a velocity fieldflux denoted by J iv. The electrical-migration flux Je of the ions is the number of moles of ionspassing through a unit area per second and is given by:J ie = µiciqiFE, (2.4)where µi is the ion mobility, ci is the concentration, qi is the charge, F is the Faraday constant,and E denotes the electric field. The diffusive flux J id produced by concentration gradients isgiven by:J id = −Di∇ci, (2.5)where Di is the diffusion coefficient. The velocity flux is given byJ iv = vci, (2.6)where v is the velocity field and will be computed from the Navier-Stokes equation for fluid flow.Using (2.4), (2.5), and (2.6) the total flux of species i is given by the Nernst-Planck equation:J i = J ie + J id + J iv= −Di∇ci + µiciqiFE + vci= −Di(∇ci − qikBTciE) + vci, (2.7)where the last relation is obtained by substitution of the Einstein relation µi = Di/kBT , wherekB is the Boltzmann constant, and T the temperature of the solution. The electrical field E,in (2.4), is obtained from Maxwell’s equations. Note that electromagnetic wave phenomenaoccur on a time scale of nano-seconds, while the time scale of interest for the electro-diffusionprocess occurs at the milli-second scale; therefore, we assume the electrostatic approximationof Maxwell’s equations applies allowing the use of Poisson’s equation to obtain an expressionfor E given by:∇ · (εE) = ρE = −∇φ=⇒ ∇ · (ε∇φ) = −ρ, (2.8)where ε is the dielectric permittivity, ρ is the charge density, and φ is the electric potential.We now impose mass conservation which states that the time change of concentration mustbe equal to the divergence of the total flux. Using mass conservation we obtain the following242.3. Dynamic Model of the Ion Channel-Switch Biosensorexpression known as the PNP theory that describes the electro-diffusion of ions in solution:∂ci∂t = −∇ · JiJ i = J id + J ie + J iv = −[Di(∇ci + qikBTci∇φ)]+ vci∇ · (ε∇φ) = −ρ = −F∑iqicifor i = {1, 2, . . . , n}. (2.9)Recall that parameter i ∈ {1, 2, . . . , n} denotes the ionic species, ci is the concentration ofspecies i, Di is the diffusivity of species i, qi is the charge of species i, F is the Faradayconstant, kB is Boltzmann’s constant, T is the temperature, ε is the dielectric permittivity,J i is the ionic flux, and φ is the potential. Note that if steric effects are important then theGeneralized PNP equation, presented in Chapter 3, can be used; however, in this Chapter stericeffects contribute negligibly to the dynamics.The concentration of dimers cds(t) is related to the analyte dynamics (2.9) by a reactionboundary condition ∂Ωsurf (Fig. 2.2) at the surface of the membrane in contact with the elec-trolyte. On ∂Ωsurf, analyte molecules bind to the tethered antibody sites followed by a cross-linking of the mobile gA monomers to the bound analytes. The primary species involved in thisprocess include the analytes a, binding sites b, mobile gA monomers c, tethered gA monomerss, and the dimers d, with respective concentrations {ca, cbs, ccs, css, cds}. Other chemical complexespresent include w, x, y, and z with concentrations {cws , cxs , cys , czs}. The chemical reactions thatrelate these chemical species are described by the following set of reactions [13–15]:a+ b f1−⇀↽−r1w a+ c f2−⇀↽−r2x w + c f3−⇀↽−r3y x+ b f4−⇀↽−r4yc+ s f5−⇀↽−r5d a+ d f6−⇀↽−r6z x+ s f7−⇀↽−r7z. (2.10)In (2.10), rj and fj , for i ∈ {1, 2, 3, 4, 5, 6, 7}, denote the reverse and forward reaction rates forthe chemical species {a, b, c, d, s, w, x, y, z}. An explanation of the reactions that take place canbe found in [15]. The dynamics of the surface bound species are modeled using a set of surfacereaction-diffusion equations given by:∂cjs∂t = ∇s · (Djs∇scj) +Rjs. (2.11)In (2.11), cjs is the surface concentration of species j ∈ {a, b, c, d, w, x, y, z}, Dis is the surfacediffusion, Rjs is the change in concentration resulting from the chemical reactions (2.10), and∇s is the surface gradient.252.3. Dynamic Model of the Ion Channel-Switch BiosensorGiven the time-scale of the conductance measurements is seconds, we assume that thevelocity field v is a fully developed laminar flow with a parabolic velocity profile given by:v(z) =( 6QLwh)( zh)(1− zh), (2.12)where Q,W, and h are defined in Fig. 2.2 and Table 2.1.Lhz-axis, n∂ca∂t = −∇ · (Ja)− v(z)∇cav(z) = ( 6QLwh)(zh)(1− zh)∂Ωin∂Ωsurf∂Ωb ∂Ωbn · Ja = 0n·Ja=0Q⇒Figure 2.2: Schematic of the ICS biosensor model (2.9), (2.10), (2.11), and (2.12) with boundaryconditions (2.13). The analyte enters the ICS at ∂Ωin with a flow rate of Q. ∂Ωsurf is the surfaceof the tethered membrane, and ∂Ωb denotes the boundary of the membrane. n denotes theinward normal vector from the surface. Other parameters are defined in Table 2.1.For uncharged analyte species zi = 0 in (2.9) which reduces the system of non-linear PDEsin (2.9) into the standard advection-diffusion PDE. The boundary and initial conditions of (2.9)and (2.11) are given by:n ·Da∇ca = Ras in ∂Ωsurf, n ·Da∇ca = 0 otherwise,ca = cao in ∂Ωin, n ·Djs∇scjs = 0 in ∂Ωb, (2.13)with ∂Ωsurf, ∂Ωin, and ∂Ωb defined in Fig. 2.2, and initial conditions ca(0) = 0 and cjs(0) definedin Table 2.1. In (2.13), Ras couples the change in analyte concentration (units of mol/m3) tothe change in surface bound species concentration (units of mol/m2) via the reactions in (2.10),and cao is the input analyte concentration.The surface concentration of gA dimers, denoted by cds , is computed from (2.9), (2.10),(2.11), and (2.12) with the boundary conditions (2.13), and is related to the membrane con-ductance by:Ĝm(t) = κ∫∂Ωsurfcds(t)dS +Go, (2.14)where Ĝm(t) is the estimated conductance, κ is the proportionality constant relating the con-ductance of the gA dimers to the number of gA dimers, with ∂Ωsurf defined in Fig. 2.2.Remark: For sufficiently high analyte concentrations, input flow rate, or low binding sitedensities with Djs (2.11) sufficiently small, then the dynamics of the membrane conductanceis in the reaction-rate limited regime in which the surface reactions govern the membraneconductance. In the reaction-rate limited regime the membrane conductance Ĝm(t) in (2.14)262.4. Experimental Measurements Ion Channel-Switch Biosensor: Streptavidin, TSH, Ferritin, hCGcan be computed from the solution of an ordinary differential equation, refer to [15] for details.The reaction-rate limited regime occurs if the characteristic time scale of the surface reactionsare less then the characteristic time scale of diffusion τ = L2/Da.2.3.3 Computing Model Parameters using Maximum Likelihood EstimatorHow can we estimate the reaction rates in (2.10) given the time-dependent conductance of theICS biosensor. From (2.9), (2.10), (2.11), and (2.12) with the boundary conditions (2.13) wecan estimate the membrane conductance Ĝm(t) (2.14) given the parametersθ = {f1, . . . , f6, r1, . . . , r6, Da, cao, D1s , . . . , D6s} with the flow rate Q and initial conditions known.From the experimental measurements we have a finite number of conductance measurementsgiven by Gm = {Gm(T1), Gm(T2), . . . , Gm(TK)}. The estimation of the model parameters inthe least-squares sense requires the solution of the following constrained optimization problem:θ∗ ∈ arg minθ∈R+{ K∑i=1(Gm(Ti)− Ĝm(Ti))2}. (2.15)In (2.15), the parameter θ∗ denotes the solution to the constrained optimization problem. Notethat typically only the diffusion and initial concentration Da, cao are not known when fitting(2.15) to experimental data. The parameter cds(i; θ) is computed using (2.9) and (2.13) usingthe parameters in θ. To estimate θ∗ we utilize the “Levenberg-Marquardt” algorithm. Given theconductance measurements of the ICS are corrupted by independent stationary white Gaussiannoise, this nonlinear least squares method provides the maximum likelihood estimate of themodel parameters θ.2.4 Experimental Measurements Ion Channel-SwitchBiosensor: Streptavidin, TSH, Ferritin, hCGIn this section the model developed in Sec. 2.3, which relates membrane conductance to analyteconcentration, is applied to experimental measurements of: streptavidin, thyroid stimulatinghormone (TSH), ferritin, and human chorionic gonadotropin (hCG). As shown, the numeri-cally predicted conductance (2.14) from the reaction-diffusion model coupled with fluid flowdynamics provide accurate predictions of the biosensor response over a wide range of analyteconcentrations.Fig. 2.3 presents the experimentally measured and numerically predicted conductance ofthe ICS biosensor for the concentration estimation of streptavidin, TSH, and ferritin. As seen,the experimentally measured conductance is in good agreement with the numerically predictedconductance. For low analyte concentrations (i.e. pM), the diffusive dynamics of the analytesignificantly influences the population of gA dimers present in the tethered membrane. As seen272.4. Experimental Measurements Ion Channel-Switch Biosensor: Streptavidin, TSH, Ferritin, hCGin Fig. 2.3, the dynamic model presented in Sec. 2.3 can account for the diffusive dynamics.This allows the dynamic model to be used for not only concentration estimation, but the designof the ICS biosensor. If a specific concentration of analyte is to be measured using the ICSbiosensor, the dynamic model presented in Sec. 2.3 can be used to select the number of bindingsites and flow rate necessary for measurement. This procedure was applied for the design of thenumber of binding sites for the detection of TSH from 100 fM-350 pM, and for the detectionof ferritin from 100 fM-100 pM. As seen in Fig. 2.3(c), using the selected number of bindingsites, mobile gA monomers, and flow rate, the concentration of TSH in the range of 100 fM-350pM can be estimated using the ICS. Fig. 2.3(d) presents the measurement of ferritin in wholeblood. As mentioned in Sec. 2.1, the ICS is designed to only detect specific target species. Asthousands of molecular compounds are present in human blood, a remarkably good estimateof ferritin can be obtained using the measured impedance of the ICS and the predictive modelpresented in Sec. 2.3.0 100 200 300 400 5000.20.40.60.81Time [s]Normalized Conductance G m(t)/Go  ExperimentalPredicted1000 pM10 pM100 pM(a) Streptividin concentrations cA: 10 pM,100 pM, 1000 pM.0 100 200 3000.20.40.60.81Time [s]Conductance Gm(t)/Go  ExperimentalPredicted200 pM400 pM600 pM(b) Ferritin in PBS, concentrations cA: 200pM, 400 pM, 600 pM.0 50 100 150 200 2500.30.40.50.60.70.80.9Time [s]Conductance Gm(t)/Go  ExperimentalPredicted100 fM312 pM2 pM(c) TSH concentrations cA: 312 pM, 2 pM,100 fM.0 100 200 3000.40.50.60.70.80.91Time [s]Conductance Gm(t)/Go  ExperimentalPredicted50 pM(d) Ferritin in whole blood, concentrations cA:0 pM, 50 pM.Figure 2.3: Experimentally measured and numerically predicted normalized conductanceGm(t)/Go for Streptavidin, TSH and Ferritin. The numerical predictions are computed us-ing the ICS biosensor model in Sec. 2.3 with the parameters defined in Table 2.1.282.4. Experimental Measurements Ion Channel-Switch Biosensor: Streptavidin, TSH, Ferritin, hCGFig. 2.4 presents the measured and numerically predicted conductance of the ICS biosensorfor the concentration estimation of hCG. For concentrations of hCG above 10 nM in blood orurine, this suggests a blastocyst or mammalian embryogenesis is present. The experimentallymeasured concentrations of hCG used are 0 nM and 353 nM. As seen from Fig. 2.4, theseproduce very different membrane conductance dynamics. This could alone be used as a testfor pregnancy; however, it is of clinical importance to know the concentration of hCG present.Using the experimentally measured impedance and the predictive model in Sec. 2.3, an accurateestimate of the concentration of hCG is possible. From Fig. 2.4, at t = 40 seconds, the changein the number of conducting gA dimers is negligible. Therefore the membrane conductance isat its equilibrium value and results because only aqueous pores are present for t > 40 s.In [14], clinical samples were used to establish that the ICS biosensor can detect influenzaA. In a clinical environment, it is of importance to estimate the concentration of airborneinfluenza A. Using the models presented in this chapter, the ICS biosensor can be designed forthis purpose.Time [s]Conductance G m(t)/G o  0 20 40 60 80 100 120 140 160 180 2000.20.40.60.81 ExperimentalPredicted0 nM10 nM353 nM 12 nMFigure 2.4: Experimentally measured and numerically predicted normalized conductanceGm(t)/Go for the human chorionic gonadotropin (hCG) concentrations cA: 0 nM, 353 nM.The numerical predictions are computed using the ICS biosensor model in Sec. 2.3 with theparameters defined in Table 2.1.2.4.1 Experimental Setup and Numerical MethodsThe governing equations (2.9), (2.10), (2.11), and (2.12) with the boundary conditions (2.13)are solved numerically with the commercially available finite element solver COMSOL 4.3a(Comsol Multiphysics, Burlington, MA). To solve the advection-diffusion (2.9) and (2.12) theCOMSOL modules Transport of Diluted Species is used. The surface reaction diffusion equation(2.11) is solved using the Weak Form Boundary PDE module. The simulation domain is meshedwith approximately 46,000 triangular elements constructed using an advancing front meshingalgorithm. The GPNP and PNP are numerically solved using the multifrontal massively par-allel sparse direct solver [107] with a variable-order variable-step-size backward differential for-mula [108]. The “Levenberg-Marquardt” algorithm (2.15) is implemented using the MATLAB292.5. Closing Remarksfunction lsqnonlin with (2.14) computed from the results of the COMSOL simulations.All experimental measurements, unless otherwise stated, were conducted at 20oC in a phos-phate buffered solution with a pH of 7.2, and a 0.15 M saline solution composed of Na+, K+,and Cl−. At this temperature the tethered membrane is in the liquid phase. A pH of 7.2 wasselected to match that typically found in the cellular cytosol of real cells. The forward andreverse reaction rates in Table 2.1 are obtained from [13–15]. In Table 2.1, notice that thecomputed diffusion coefficient for Ferritin is 80 µm2/s and that for hCG is 250 µm2/s–this is anexpected result as the Ferritin has a molecular weight of 450 kDa and hCG a molecular weightof 25.7 kDa.Table 2.1: Model Parameters for ICS BiosensorSymbol Definition ValueLw Width of flow chamber 3.0 mmL Length of flow chamber 0.7 mmh Height of flow chamber 100 µmDjs Surface bound diffusion 0-12 µm2/scws (0), cxs(0), cys(0), czs(0) initial surface concentration 0 mol/m2Streptividin Ferritin TSH hCGcAo Inlet analyte concentration 10 pM-1 nM 50-600 pM 100 fM-312 pM 10-353 nMccs(0) Mobile gA monomers 16 pmol/m2 16 pmol/m2 16 pmol/m2 1.6 pmol/m2css(0) Tethered gA monomers 166 pmol/m2 166 pmol/m2 16 pmol/m2 16 pmol/m2cbs(0) Tethered binding sites 16 pmol/m2 16 pmol/m2 33 pmol/m2 33 pmol/m2cds(0) gA dimers 16 pmol/m2 16 pmol/m2 33 pmol/m2 33 pmol/m2f1 = f2 = f6 Forward reaction rate 4×103 m3/smol 4×103 m3/smol 8×103 m3/smol 5×102 m3/smolf3 = f4 Forward reaction rate 3×1011 m2/smol 3×1011 m2/smol 3×1011 m2/smol 3×109 m2/smolf5 = f7 Forward reaction rate 6×109 m2/smol 6×109 m2/smol 6×108 m2/smol 3×1011 m2/smolr1 = r2 = r6 Reverse reaction rate 10−6 s−1 10−6 s−1 10−6 s−1 10−4 s−1r3 = r4 Reverse reaction rate 10−6 s−1 10−6 s−1 10−6 s−1 10−4 s−1r5 = r7 Reverse reaction rate 1.5×10−2 s−1 1.5×10−2 s−1 1.5×10−2 s−1 1.0×10−2 s−1Da Diffusivity of analyte a 150 µm2/s 80 µm2/s 150 µm2/s 250 µm2/sQ Flow rate 100 µL/min 10 µL/min 100 µL/min 100 µL/min2.5 Closing RemarksIn this chapter we have presented the formation, dynamic models, and experimental measure-ments using the ICS biosensor. Key features include using a lumped circuit model for thebioelectronic interface, electrolyte, and tethered membrane allowing the conductance of themembrane to be estimated from current measurements. The membrane conductance is relatedto the population of conducting gramicidin dimers. The population of conducting dimers isdependent on the electrodiffusive behaviour of the target molecule of interest and the chemicalreactions at the tethered membrane surface. Employing an advection-diffusion partial differ-ential equation coupled with nonlinear interface boundary conditions, we illustrated how theconcentration of specific analyte molecules can be computed using a maximum-likelihood esti-mator which simultaneously estimated the surface reaction rates of the ICS. Given the dynamicmodels presented in this work coupled with engineered receptors allows the ICS to be used for302.5. Closing Remarksa host of applications for rapid detection of specific analyte molecules in cluttered electrolyteenvironments.313Pore Formation MeasurementPlatform3.1 IntroductionThis chapter presents an atomistic-to-observable model for the pore formation dynamics of theantimicrobial peptide PGLa in a tethered membrane. Gaining insight into the pore forma-tion dynamics of peptides and proteins in tethered membranes is crucial for the developmentof novel drugs, gene delivery therapies, and controlling pore formation in cell-like bioreactors.The proposed model consists of coarse-grained molecular dynamics, a continuum model com-posed of a generalized version of Fick’s law of diffusion coupled with surface reaction-diffusionequations, and a fractional order macroscopic model. We validate the model using experimen-tal measurements of the pore forming toxin α-Hemolysin produced by Staphylococcus aureus.Using experimental measurements and the atomistic-to-observable model the dynamics andself-assembly process of PGLa pore formation in charged and uncharged membranes whichmimic prokaryotic and eukaryotic membranes are provided. We show that PGLa dimers canbind to the membrane surface, and enter the transmembrane configuration via transient aque-ous pores. It is also illustrated how PGLa can bind, enter the transmembrane configuration,and oligomerize once in the transmembrane state.Pore forming peptides and proteins are crucial to the attack and defense mechanisms ofbiological organisms. Understanding the chemical kinetics of pore forming peptides and pro-teins provides vital information of use to pharmacologists to target specific classes of pep-tides/proteins for in depth pharmaceutical screening of novel drugs. In this chapter we studythe pore formation dynamics of PGLa (peptidyl-glycineleucine-carboxyamide), a membrane-active antimicrobial peptide produced in specialized neuroepithelial cells in the African frogXenopus laevis [109]. In addition to the antimicrobial activity of PGLa, it also contains an-ticancer [110, 111], antiviral [112], and antifungal properties [113]. The remarkable featureof PGLa is that it provides a potential source for new antibiotics against increasingly com-mon multiresistant pathogens (i.e. “superbugs”) such as methicillin-resistant Staphylococcusaureus [114].323.1. IntroductionUsing 2H-, 15N-, and 19F-NMR spectroscopy the topology of PGLa in membranes has beenobserved which provide insight into the intermediate states of PGLa leading to pore forma-tion [115–118]. PGLa is in a α-helical conformation when membrane bound. The orientationof the membrane bound PGLa is dependent on the lipid composition, peptide/lipid ratio, hy-dration, temperature, and the pH [115]. From molecular dynamics [119] and 19F-NMR [117]the long axis of PGLa is shown to be aligned parallel to the membrane surface in a monomericstate for peptide to lipid concentrations below 1:200. However for high concentrations (≥1:50) a tilted dimerization state is observed. The transmembrane state of PGLa has only beenobserved at temperatures below 15 ◦C when the membrane is in the gel phase [116]. WhenPGLa is in the transmembrane state, an oligomerization process is suggested which leads to theformation of conducing toroidal pores. Using the observed NMR topologies and orientation,the reaction mechanism for PGLa pore formation is suggested to involve PGLa binding to themembrane, translocation via transient aqueous pores to the transmembrane state, and finallyoligomerization to form conducting pores.The above experimental studies motivate the development of predictive models for the dy-namics of pore formation. It is crucial for such modeling to include tethering in the structureof the cell membranes. Tethers are essential in mimicking real biological membranes. In thischapter an atomistic-to-observable model and experimental measurements from a Pore Forma-tion Measurement Platform (PFMP) are used to investigate the pore formation dynamics ofPGLa. The tethered membrane platform was developed by Cornell [10], with the constructionand operation provided in [3, 16]. A schematic of the PFMP is provided in Fig. 3.1. The PFMPis composed of a self-assembled monolayer of mobile lipids, and a second layer comprising aself-assembled monolayer of tethered and mobile lipids. The tethered lipids are anchored to thegold electrode via polyethylene glycol chains. As pores are formed in the membrane, the conduc-tance of the membrane increases. This increase in conductance can be detected by measuringthe current response of the PFMP resulting from an applied potential across the electrodes. Toextract pore formation dynamics from the PFMP requires the utilization of a dynamic model.The model must link the diffusive behavior of PGLa in solution to the surface reactions that in-clude effects from binding, translocation to the transmembrane state, oligomerization, and poreclosing if present. In this chapter an atomistic-to-observable model that comprises three levelsof abstraction: fractional order macroscopic model, a generalized reaction-diffusion continuummodel, and coarse-grained molecular dynamics (CGMD) is proposed.Developing the proposed atomistic-to-observable model requires careful interfacing betweenthe three levels of abstraction. We achieve this interfacing as follows (see Fig. 3.2 for a schematicillustration). The CGMD model is used to estimate the diffusion coefficient D of surface boundPGLa peptides and gain insight into the dynamics of binding, translocation, and oligomerizationrequired for pore PGLa pore formation. The computed diffusion coefficient is then used in ageneralized reaction-diffusion model which accounts for the steric effects of large molecules using333.1. IntroductionAqueousPoreGold Electrode−⇀↽−Surface Binding −⇀↽− −⇀↽−OligomerizationGp I(t)Figure 3.1: Schematic of the Pore Formation Measurement Platform (PFMP). A voltage po-tential is applied between the gold electrode and counter electrode (not shown) and the currentresponse I(t) is measured. The PGLa peptide binds to the membrane surface, then undergoesoligomerization steps to create a PGLa pore with conductance Gp. The current response I(t) ofthe PFMP is dependent on the number of conducting PGLa pores and the equilibrium numberof aqueous pores present in the tethered membrane.a “Langmuir” like activity coefficient. The generalized reaction-diffusion equation is coupled tothe surface reaction equations via a “Langmuir-Hinshelwood” like equation classically used todescribe surface binding of molecules. As conducting pores form in the tethered membrane, theconductance of the membrane will increase proportionally to the number of pores. This allowsthe concentration of surface bound pores to be used to estimate the membrane conductanceĜm(t). Current measurements from the PFMP are used in the fractional order macroscopicmodel to estimate the membrane conductance Gm(t). The final step to validate the poreformation reaction mechanism of PGLa is to compare the numerically estimated conductanceĜm(t) with measured conductance Gm(t) from the PFMP.3.1.1 Main Results and Chapter OrganizationThe formation and operation of the PFMP is provided in Sec. 3.2, and dynamic model is pro-vided in Sec. 3.3. The novelty of the proposed model compared to those in [3, 16] is threefold:First the diffusion coefficients are computed from coarse-grained molecular dynamics, seconda generalized reaction-diffusion equation is used to model the pore formation, and thirdly afractional order macroscopic model is used to account for the diffusion-limited charge trans-fer, reaction limited charge transfer, and ionic adsorption dynamics at the bioelectronic inter-face of the PFMP. Prior to investigating PGLa pore formation, we validated the PFMP andatomistic-to-observable model by studying the pore formation dynamics of the protein toxinα-Hemolysin [94]. α-Hemolysin is secreted from Staphylococcus aureus and binds with humanplatelets, erythrocytes, monocytes, lymphocytes, and endothelial cells ultimately causing celldeath. We confirm the pore formation dynamics of α-Hemolysin [120, 121] using experimental343.1. IntroductionCoarse-GrainedMolecular DynamicsGeneralizedReaction-DiffusionPore FormationMeasurement Platform Fractional Order ModelDGˆm(t)I(t)Gm(t)Gm(t)Gˆm(t)0 1,000 2,000 3,000 4,000 5,000024Conductance[µS]Time [s]Figure 3.2: Schematic of the atomistic-to-observable model. D is the diffusion coefficient ofbound PGLa peptides, Ĝm(t) is the predicted conductance, I(t) is the measured current fromthe PFMP (Fig. 3.1), and Gm(t) is the measured conductance.measurements from the PFMP and atomistic-to-observable model.Having validated the PFMP and model, in Sec. 3.4 the pore forming reaction pathwayof PGLa is estimated in charged and uncharged membrane surfaces that mimic prokaryotic,eukaryotic, and archaebacterial membranes. The key findings of the reaction mechanism ofPGLa include:1. At high peptide to lipid concentrations above 1:200 PGLa dimers can bind to the mem-brane surface providing an explanation for the observed dimer structures experimentallymeasured using NMR spectroscopy [115–118, 122].2. The transition from the surface bound conformation to the transmembrane conformationis facilitated by transient aqueous pores. The resulting transmembrane conformation isin agreement with the NMR spectroscopic measurements reported in [116, 123].3. The oligomerization of PGLa can occur when in the transmembrane conformation as aresult of the amine-terminus or carboxyl-terminus of PGLa monomers interacting.4. Using experimental measurements from the PFMP the potency of PGLa for killing nega-tively charged membranes that are typically found in prokaryotes is illustrated. We showthat PGLa not only increases the number of pores in negatively charged membranes, butthe lifetime of conducting pores also increases compared to the lifetime of PGLa pores inuncharged membranes.Although we only apply the PFMP and atomistic-to-observable model to the study of poreformation dynamics of PGLa, the platform and modeling methodology are general and can beused for other pore forming peptides and proteins of interest. Closing remarks are provided inSec. 3.5.353.2. Pore Formation Measurement Platform: Construction and Formation3.2 Pore Formation Measurement Platform: Construction andFormationThe tethered membrane is supported by a 25×75×1 mm polycarbonate slide. Six 100 nmthick sputtered gold electrodes, each with dimensions 0.7×3 mm, rest on the polycarbonateslide. Each electrode is in an isolated flow cell with a common gold return electrode. Theformation of the tethered membrane on the gold electrode is performed in two stages using thesolvent-exchange technique presented in [10]. The first stage of formation involves anchoringof the tethers and spacers to the gold surface. The tethers provide structural integrity to themembrane and mimic the physiological response of the cytoskeletal supports of real cell mem-branes. The spacers laterally separate the tethers allowing patches of mobile lipids to diffusein the membrane. The spacer is composed of a benzyl disulphide connected to a four-oxygen-ethylene-glycol group terminated by an OH; the tethers are composed of a benzyl disulphideconnected to an eight-oxygen-ethylene-glycol group terminated by a C20 hydrophobic phytanylchain. To form the anchoring layer, an ethanolic solution containing 370 µM of engineeredratios of benzyl disulphide components is prepared. This solution is exposed to the gold surfacefor 30 min, then the surface is flushed with ethanol and air dried for approximately 2 min.Note that in the special case of 100% tethering, the engineered tethered membrane is com-posed of a tethered archaebacterial based monolayer with no spacer molecules. Stage 2 involvesthe formation of the tethered membrane. The tethered membrane mobile lipids are composedof engineered mixtures of zwittrionic C20 diphytanyl-either-glycero-phosphatidylcholine lipid(DphPC), and C20 diphytanyl-diglyceride either (GDPE). A solution containing a mixture ofmobile lipids is brought into contact with the gold bonded components from Stage 1. Severallipid solvents can be used [9, 10], however in most cases the lipids selected to form the bilayerare soluble in ethanol. As an example, let us consider the formation of a 70% DphPC and 30%GDPE mixed tethered membrane. 8 µL of 3 mM of the 70% DphPC and 30% GDPE ethanolicsolution is added to the flow chamber. The solution is incubated for 2 min at 20oC in whichthe tethered membrane forms. Following the 2 min incubation, 300 µL of phosphate bufferedsaline is flushed through each flow chamber. The tethered membrane is equilibrated for 30 minprior to performing any experimental measurements. The formation of the Escherichia coli,and Saccharomyces cerevisiae tethered membranes follows a similar procedure and is thereforeomitted.3.3 Dynamic ModelIn this section the atomistic-to-observable model for the study of the antimicrobial peptidePGLa is provided. Given the initial concentration of PGLa, the model is used to predict thetethered membrane conductance of the PFMP. If the predicted conductance and experimen-363.3. Dynamic Modeltally measured conductance from the PFMP are in good agreement then the model accuratelypredicts the reaction pathway for PGLa pore formation (refer to Fig. 3.2).3.3.1 Fractional Order Macroscopic ModelThe fractional order macroscopic model of the PFMP is identical to the fractional order macro-scopic model for the ICS biosensor provided in Fig. 2.1, however for the PFMP the membraneconductance Gm is dependent on the number of aqeous conducting pores Go and the populationof conducting PGLa pores. Using a sinusoidal drive potential Vs(t) = Vo sin(2pift) with fre-quency f and magnitude Vo below 50 mV, the current response of the PFMP can be computedusingI(f) = Vo[Re +1Gm + j2pifCm+ 1(j2pif)pCdl]−1, (3.1)where j denotes the complex number√−1, and the parameters Re, Cm, Cdl are defined in (2.3).The associated impedance of the PFMP, denoted by Z(f), is given by the content inside [·] in(3.1). Assuming Gm(t) is static during the measurement of I(f), the membrane conductanceGm(t) can be computed using a least-squares estimator with a cost function given by thedifference between the measured current and the computed current from (3.1).3.3.2 Generalized Reaction-Diffusion Continuum ModelTo link the surface diffusion D obtained from coarse-grained molecular dynamics with theexperimentally measured conductance Gm(t) from the fractional order model, we constructa generalized reaction-diffusion continuum model. This continuum model accounts for thechemical reactions leading to PGLa pore formation, diffusion of PGLa to the membrane surface,and the surface diffusion of PGLa.Surface Reaction of PGLaThe chemical reactions leading to pore formation occur at the surface of the tethered membrane,denoted by ∂Ωsurf. The conductance of the membrane Gm(t) is dependent on the concentrationof conducting pores in the membrane. To compute the concentration of conducting pores weconsider two possible reaction mechanisms all which contain a surface binding step, oligomer-ization, pore formation, and finally pore closure.Reaction Mechanism 1: The first reaction mechanism we consider is based on those presentedin [124–126] in which soluble peptides bind to the surface of the membrane, then rapidly forma conducting pore. We also include the possibility of pore closure as the lifetime of pores isfinite. The reaction mechanism we consider is given byak1a−⇀↽−kdm1kp−→ p1 np1 k1−→ pn mpn kc−→ c, (3.2)373.3. Dynamic Modelwhere m1 is the membrane bound monomer, p1 is the protomer, pn is the conducting porecontaining n protomers, and c is a closed pore. Note that after the peptide binds to thesurface, it may undergo conformational and/or orientation changes prior to forming the pore.Once these changes have completed the monomer is denoted as a protomer. In (3.2) kd is thedissociation constant, kp is the rate of protomer formation, k1a the association rate constant, k1rate of protomer binding which includes the translocation of the peptide from the surface to thetransmembrane orientation, and kc the rate of pore closing with n andm denoting stoichiometricnumbers. We define the association rate constant k1a as decreasing as the number of membranebound peptides increases. If we denote mmax as the maximum number of bound peptides, thenthe association constant is defined as k1a = ka(mmax−m1−p1−npn−nmc). Note that if a wasconstant, then the surface binding mechanism in (3.2) resembles the “Langmuir-Hinshelwood”equation that is classically used to describe the dynamics of adsorption process at surfaces.Remark: The reaction mechanism (3.2) is a Hill-type approximation [127] of the aggregationand binding processes of the reaction mechanisms presented in [128] for α-Hemolysin pores, andin [126] for Cytolysin A pores. We do not consider the sequential binding as a result of modelidentifiability–that is, if (3.2) is in agreement with the experimentally measured results, thena sequential binding process is guaranteed to fit the data with a series of slow binding stepsfollowed by a fast binding step.Reaction Mechanism 2: The second reaction mechanism we consider is that a dimerizationstep is required prior to peptides binding to the surface of the membrane at high peptide con-centrations. Using the CGMD model we show that PGLa dimers can insert into the membraneto form a membrane bound dimer. The membrane bound PGLa dimer has been experimentallymeasured using 2H-, 15N-, and 19F-NMR spectroscopy [115–118] for high peptide concentra-tions. The associated reaction mechanism we consider is given by:2ak2a−⇀↽−kdm2kp−→ p2 np2 k2−→ p2n mp2n kc−→ cfor i = 1, 2, . . . , n/2− 1, (3.3)where m2 is the bound dimer, p2 are protomer dimers, p2n is a conducting pore with n dimers,and k2 is the rate of protomer dimer formation. In (3.3) the association rate constant is givenby k2a = ka(mmax − 2m2 − 2p2 − 2np2n − 2nmc).Diffusion of PGLaWe now consider the second level of the three level model. The concentration of conductingPGLa pores is dependent on the chemical reactions on the membrane surface and on the dif-fusion dynamics of PGLa. To model the diffusion dynamics of PGLa in solution we utilize a383.3. Dynamic Modelgeneralized version of Fick’s law given by:∂a∂t = ∇ ·(Da∇a+Daa∇ ln(1−NAr3aa)), (3.4)where a is the concentration of PGLa in solution, Da is the diffusion coefficient of PGLa, NAis the Avogadro’s constant, and ra is the effective radius of PGLa in solution. The maximumconcentration of analyte possible is given by amax = 1/NAr3a which assumes a cubic packingstructure. Note that for (3.4) to be a suitable model, electrodiffusive effects must be negligiblein the electrolyte. This assumptions holds for Fqauaa∇φ 1, where F is Faraday’s constant,qa is the charge of PGLa, and ua is the ionic mobility PGLa. The generalized version of Fick’slaw includes a “Langmuir” type activity coefficient to account for the steric effects of PGLa–that is, the steric effects are accounted for by modifying the associated concentration. As seen,if ra = 0 then we obtain the standard Fick’s law of diffusion from (3.4).In the membrane there exists surface bound PGLa with concentrationm, protomer monomers,protomer dimers, higher order protomer complexes, and conducting pores, with concentrationsgiven by: a,m, p1, p2, . . . , pn. The dynamics of the PGLa peptide complexes in the membraneare governed by the following surface reaction-diffusion partial differential equations:∂m∂t = ∇s · Jm +Rm,Jm = (Dm∇sm+Dmm∇ ln(1−NA(r2mm+∑r2i pi)),∂pi∂t = ∇s · Ji +Ri,Ji = (Di∇spi +Dipi∇ ln(1−NA(r2mm+∑r2i pi)),for i ∈ {1, 2, . . . , n}. (3.5)In (3.5) ∇s is the surface gradient, D is the surface diffusion coefficient of the respective species,r is the effective radius of each species, and Rm denotes the change in concentration as a resultof PGLa binding to the membrane surface, and Ri the subsequent chemical reactions leadingto pore formation. The boundary conditions of (3.4) and (3.5) are given by:n ·Da∇a = Ra in ∂Ωsurf, n ·Da∇a = 0 otherwise ,n ·Dm∇sm = 0 in ∂Ωb, n ·Di∇spi = 0 in ∂Ωb. (3.6)with ∂Ωsurf the membrane surface, and ∂Ωb the boundary of the membrane surface, and n theunit normal vector. In (3.6), Ra denotes the binding process of the PGLa peptide in solutionto the membrane bound state m. The chemical reaction rates Ra, Rm, and Ri can be computedfrom (3.2) and (3.3) for each of the two reaction mechanisms. Initially the solution of PGLawith concentration ao is inserted into a flow-cell chamber defined by ∂Ωin (Fig. 3.13). The393.3. Dynamic Modelinitial conditions of (3.4) and (3.5) are given by:a|t=0 = ao in ∂Ωin, a|t=0 = 0 otherwise ,m|t=0 = 0 in ∂Ωsurf, pi|t=0 = 0 in ∂Ωsurf. (3.7)The conductance of the membrane is dependent on the concentration of conducting PGLapores and the equilibrium number of aqueous pores in the membrane resulting from randomthermal fluctuations. Denoting Go as the conductance of the aqueous pores, then the totalconductance of the membrane is given by:Ĝm(t) = Go + κp∫∂Ωsurfpn(t, x)dS. (3.8)In (3.8), κp, with units of S/mol, is a proportionality constant relating the mean conductanceof the pores to the molar concentration of pores pn. The mean conductance of each PGLapore Gp (Fig. 3.1) is equal to κp/NA where NA is Avagadro’s constant. From experimentalmeasurements and theory Gp is expected to be in the range of pS-nS [129, 130].Given the initial concentration ao, the diffusion coefficients D from coarse-grained moleculardynamics, the governing equations (3.4) and (3.5) with the boundary conditions (3.6) andinitial conditions (3.7), the conductance of the membrane can be estimated using (3.8). Givenmeasurements, the experimental membrane conductance denotedGm(t), can be evaluated as thesolution of the fractional order macroscopic model. Then least squares estimates of the reactionrate constants Ra(t), Rm(t), Ri(t) that describe the pore formation reaction mechanism can beobtained by minimizing (Gm(t)− Ĝm(t))2 at each time t.3.3.3 Coarse-Grained Molecular Dynamics of PGLaCGMD constitutes the third level of our three level model. This section presents the CGMDmodel of the interaction of PGLa with a DphPC membrane. The CGMD model is used to gainkey insights into the reaction mechanism and dynamics leading to PGLa pore formation, and thediffusion coefficients D in (3.5). Given the system size and the requirement to reach simulationtime horizons of several hundred nanoseconds, we employ the MARTINI force-field [75, 76] toperform all molecular dynamics simulations.The antimicrobial peptide PGLa contains 21-residues with with amino-acid sequence(GMASKAGAIAGKIAKVALKAL-NH2). As a result of recent advances in 2H-, 15N-, and 19FNMR spectroscopy [119, 123, 131–135] it is known the PGLa peptide has a α-helical configu-ration when membrane bound. The transmembrane state of PGLa (i.e. the long axis of thepeptide is parallel to the membrane normal) has not been observed at physiological temperatureswith NMR as the lifetime of conducting pores is too short for NMR measurement [116, 122, 123].403.3. Dynamic ModelHowever from recent NMR measurements [123] when a 1:1 mixture of PGLa and Maginin 2 isused, the PGLa has been observed in the transmembrane state and retains the α-helical struc-ture. Though the transmembrane configuration of PGLa, with no additional peptide present,has not been observed experimentally, it is suggested that for high peptide to lipid ratios theconfiguration of the transmembrane PGLa monomer retains the α-helical structure [122], acommon trait of similar α-helical peptides including Alamethicin [136], Maginin 2 [137], andMelittin [138]. Additionally it is known that amine-terminus and carboxyl-terminus of antimi-crobial peptides are thermodynamically stable (i.e. local energy minimum) when in contactwith the top and bottom surface of the membrane [139]. Therefore, for all CGMD simulationsthe secondary structure of the PGLa is constrained to have a α-helical structure.The all-atom PGLa is constructed using the software Molefacture contained in VMD [140].The secondary-structure of the membrane bound PGLa is defined by a α-helix with φ = −57oand ψ = −47o. The all-atom PGLa is coarse-grained for use with the MARTINI force fieldusing the protocol described in [2] with each CGMD bead representing approximately fouratoms. A schematic of the all atom structure of PGLa and coarse-grained PGLa structure areprovided in Fig. 3.3. The membrane is modeled using 512 DphPC CGMD molecules. Theparameters of the CGMD model and setup for the surface binding, translocation of surfacebound to transmembrane bound, and oligomerization are provided in Sec. 3.4.4.MARTINI Coarse-GrainingFigure 3.3: Schematic of the all-atom structure of PGLa (GMASKAGAIAGKIAKVALKAL-NH2) and the corresponding MARTINI coarse-grained structure constructed using the protocolin [2]. The PGLa backbone beads are displayed in red, and side chain beads in yellow.413.4. Numerical and Experimental Results3.4 Numerical and Experimental ResultsIn this section we estimate the chemical kinetics of PGLa pore formation using the PFMP andatomistic-to-observable model constructed in Sec. 3.3.3.4.1 Diffusion of PGLa from Coarse-Grained Molecular Dynamics(CGMD)The surface bound and transmembrane diffusion coefficients of PGLa play a central role in thedynamics of PGLa pore formation in biological membranes (refer to (3.4) and (3.5)). To esti-mate these important parameters we use the CGMD model of PGLa. The diffusion coefficientsfor surface bound and transmembrane bound monomers, dimers, and trimers is provided inTable 3.1. As expected the diffusion coefficient of the PGLa protomers decrease as the num-ber of monomers in each protomer increases. Interestingly the diffusion coefficients for thetransmembrane protomers satisfy the “free-drain limit” [141] in which the diffusion coefficientsatisfies Dn = Di/n within the error bounds. This effect has been observed for membranebound proteins using single-molecule fluorescence spectroscopy techniques [142]. Note that forprotomers that contain more then three monomers rapidly dissociate preventing an accurateestimate of the associated diffusion coefficient. The diffusion coefficient of the DphPC lipidsis 84.9 ± 0.2 µm2/s, in excellent agreement with the expected range of 10-100 µm2/s [143].The estimated diffusion of water is 2.00 ± 0.02 nm2/ns, in agreement with the experimentallymeasured diffusion coefficient of water 2.30 nm2/ns [144].Table 3.1: Diffusion Coefficients of PGLa Protomers (µm2/s)Monomer Dimer TrimerTransmembrane 91.2± 19.5 41.8± 16.0 26.25± 9.0Surface Bound 127.5± 12.4 50.7± 14.4 21.0± 8.03.4.2 Surface Binding and Oligomerization of PGLa from MolecularDynamicsIn this section the CGMD model of PGLa is used to gain key insights into the mechanismof surface binding, translocation of surface bound peptides to the transmembrane state, andoligomerization. We show that PGLa dimers can bind to the membrane surface, and thatthe transition from the surface bound state to transmembrane state is facilitated by transientaqueous pores. Surface binding and entry into the transmembrane state are key steps in thepore formation reaction mechanism in the continuum theory models presented in this chapter.Fig. 3.4 presents two mechanisms for the binding of PGLa from the bulk solution to theDphPC membrane surface. Following the method presented in [145] for monomer insertion, the423.4. Numerical and Experimental ResultsPGLa monomer is initially placed above the DphPC membrane with an α-helix configuration.Note that the membrane bound second-structure of PGLa is α-helical as measured experi-mentally using NMR spectroscopy [115–118, 122]. The peptide to lipid ratio of the monomerbinding is 1:512. At this low peptide to lipid concentration the PGLa peptide is not expectedto exist in any oligomeric state as measured experimentally using NMR measurements [115–118, 122] for peptide to lipid ratios below 1:200. As seen in Fig. 3.4 the amine-terminus ofthe PGLa monomer first binds to the surface of the DphPC membrane. At 20 ns the PGLamonomer pivots on the amine-terminus and begins to embed itself into the membrane surface.The final surface bound structure of the PGLa monomer is reached at 35 ns with the chargedlysine residues pointing into the bulk electrolyte and the hydrophobic region in contact with thehydrophobic phytanyl tails of the DphPC membrane. The monomer remains in the membraneuntil the simulation horizon is reached at 1 µs. The computed tilt angle, defined as the anglebetween the helix long-axis vector and the membrane normal, of the PGLa monomer is 90±5◦which is in excellent agreement with the 2H-NMR results of approximately 95◦ [146] and theresults from molecular dynamics [145].At high concentrations, the proximity of PGLa monomers may cause dimerization at themembrane surface prior to binding to the membrane surface. This PGLa dimer then bindsto the membrane surface. Fig. 3.4 provides snapshots of how the binding of PGLa to themembrane surface can occur for both the monomers and dimers. At 15 ns a symmetric andantiparallel PGLa dimer forms with the N-terminus of one of the PGLa monomers in the dimerinteracting with the surface of the DphPC membrane. This is an expected interaction if weconsider the structure of PGLa which has small Alanine and Glycine on one face of the helix,and large hydrophobic residues including Isoleucine and Valine on the other face. The smallresidues allow the PGLa monomers to assemble into a tightly packed dimer. The symmetricand antiparallel dimer is consistent with the Alinie and Glycine residues in close contact andthe hydrophobic residues oriented into the bulk electrolyte. At 30 ns the PGLa dimer pivots onthe amine-terminus of the PGLa and begins to embed itself into the DphPC membrane. Thedimer reaches the membrane bound state at approximately 60 ns and remains for the rest ofthe simulation horizon. The symmetric and antiparallel surface bound dimer has been observedexperimentally from NMR measurements for high peptide to lipid ratios above 1:100 [115–118, 122]. Comparing the results for surface bound state of the monomer and dimer in Fig. 3.4,we see that the dimer causes the tilt angle of the PGLa to increase compared to that of the PGLamonomer, in agreement with the experimental results from NMR measurements [115–118, 122]and the results from molecular dynamics [145].Dynamics of Pore Formation A necessary step for PGLa pore formation is the translocationof the surface bound monomers and dimers to the transmembrane state in the membrane. Howdo the peptides transition from the surface bound state to the transmembrane state? Fora 1 µs CGMD simulation we did not observe the surface bound monomer or dimer (Fig. 3.4)433.4. Numerical and Experimental Results0 ns 10 ns 35 nsMonomer Binding0 ns 30 ns 60 nsDimer BindingFigure 3.4: Snapshots of CGMD bead positions for simulation run from 0 ns to 60 ns fora DphPC membrane for the monomer binding and dimer binding of PGLa into a DphPCmembrane. The NC3 bead is displayed in blue, the PO4 bead in orange, the lipid tail carbonsin green beads, the PGLa backbone beads in red, and PGLa side chains using yellow beads.transition to the transmembrane state. The reason this occurs is that in the CGMD model, andin molecular dynamics models [145], the surface bound conformation is a thermodynamicallystable conformation. As shown in [139] using potential of mean force computations for similarlength (19 residues) α-helical peptides, the transition from the surface bound state to thetransmembrane state is not energetically favored. If however, a transient aqueous pore existsin the membrane, then the PGLa can diffuse into the transient pore. A population of transientaqueous pores exist in all membranes as a result of random thermal fluctuations.Using CGMD, Fig. 3.5 illustrates the insertion of PGLa from the surface bound state tothe transmembrane state via a transient aqueous pore. The transient aqueous pore is formedby applying a negative pressure in the lateral direction of the membrane. The PGLa monomerand dimer then diffuse into the walls of the aqueous pores as seen in Fig. 3.5. The lateralpressure is then set to 1 bar and the transient pore closes with the final PGLa monomer anddimer in the transmembrane state as seen in Fig. 3.5. The monomer and dimer remain in thetransmembrane state for the remainder of the simulation. As expected from the results in [139],the PGLa is in a thermodynamically stable conformation when transmembrane bound. Thetransmembrane structure of PGLa has not been observed by NMR spectroscopic at physiologicaltemperatures possibly as a result of the PGLa pores being transient [146]. In a gel-phaseDMPC/DMPG bilayer at temperatures below 15 ◦C, NMR measurements show that PGLa isin a transmembrane state with a tilt angle of approximately 180◦ [116]. This is in agreement443.4. Numerical and Experimental Resultswith the tilt angle of the transmembrane PGLa monomer which has a tilt angle of 168±5◦–thedifference in angle is a result of the gel-phase DMPC/DMPG bilayer having a larger membranethickness then the DphPC membrane which is in the liquid phase. Using DMPC/DMPGin a 1:3 ratio the transmembrane state of PGLa with Maginin 2 has been observed usingNMR spectroscopy for peptide to lipid ratios of 1:50 [123]. The tilt angle of PGLa from theNMR measurements in [123] is approximately 158◦, which is in agreement with the numericallycomputed tilt angle of the PGLa dimer in Fig. 3.5 of 156±3◦. This suggests that at highpeptide to lipid ratios the mixture of PGLa and Maginin 2 there exist PGLa dimers in thetransmembrane state.Side ClosedMonomer InsertionSide ClosedDimer InsertionFigure 3.5: Snapshots of the translocation of the bound monomer and dimer illustrated inFig. 3.4 into the transmembrane state via a transient hydrophilic pore. To illustrate the struc-ture of the hydrophilic pore both the top and side views as provided, the final structure displayedis the closed aqueous pore with the monomer and dimer in the transmembrane state. The col-oring scheme is identical to that used in Fig. 3.4 with water illustrated using the translucentblue beads.It has been postulated that the formation of PGLa pores may involve an oligomerizationprocess. From [145] it is known that surface bound PGLa monomers can form dimers. Is itpossible for PGLa peptides in the transmembrane state to oligomerize in the DphPC membrane?To study if the transmembrane PGLa oligomerize we initially setup a DphPC membrane with 4embedded PGLa peptides as illustrated in Fig. 3.6 with a peptide to lipid ratio of 1:128. Initiallythe monomers diffuse in the membrane. As time progresses the monomers form transmembranedimers. The formation of the transmembrane dimers from the monomers are dependent on theorientation and diffusion dynamics of the peptides, as such the first dimer is formed at 50ns, and the second at 90 ns. The formation of the two dimers occur as a result of the amine-453.4. Numerical and Experimental Resultsterminus or carboxyl-terminus interacting when two peptide come into close contact. This eventis illustrated in Fig. 3.6 which occurs at 55 ns. As seen at 200 ns, the four PGLa monomers haveformed two transmembrane dimers which remain stable for the rest of the simulation. Thereforetransmembrane monomers can oligomerize to form dimers, however at transmembrane to lipidconcentrations below 1:128 the formation of such dimers is expected to take longer as theoccurrence of peptides interacting decreases.0 ns 55 ns 200 nsFigure 3.6: Snapshots of the oligomerization of four PGLa monomers in the transmembranestate (Fig. 3.5). The initial position of the monomers is provided at 0 ns. At 55 ns a snapshotfrom the side view looking into the membrane is provided. At 200 ns the two formed transmem-brane dimers. The coloring scheme is identical to that used in Fig. 3.4 with water illustratedusing the translucent blue beads.3.4.3 Experimentally Measured Pore Formation Dynamics of PGLaIn this section we use experimental measurements from the PFMP and three level model toestimate which reaction mechanism, either (3.2) or (3.3), describe the pore formation reactionmechanism of PGLa. The PFMP contains a 10% tethering density which matches the physiolog-ical tether density of biological cells. Details on the experimental measurements and numericalmethods are provided in Sec. 3.4.4. Prior to studying the pore formation dynamics of PGLa,we validated the PFMP and atomistic-to-observable model using experimental measurementsof the pore formation dynamics of α-Hemolysin. The results are provided in Sec. 3.4.4.Prior to all experimental measurements the impedance of the tethered membrane is mea-sured. This allows us to detect if the membrane contains significant defects. Possible membranedefects include patches with the gold electrode directly exposed to the bulk electrolyte, or withportions of bilayer sandwiched together. The defect density in the membrane can be estimatedfrom the impedance measurements using the protocol presented in [102]. If the estimated ca-pacitance of the membrane significantly increases this indicates that either electrodesorption ofthe tethers and spacers has occurred, or portions of the tethered membrane have been releasedinto the electrolyte. If the estimated equilibrium aqueous pore conductance significantly in-creases then catastrophic voltage breakdown of the membrane has occurred causing separatedareas of the membrane to degrade. Typical values for membrane capacitance and conduc-463.4. Numerical and Experimental Resultstance are 0.5-1.3 µF/cm2 and 0.5-2.0 µS for an intact 10% tethered membrane with surfacearea 2.1 mm2. The estimated fractional order parameter lies in the range between 0.8 and0.9; therefore, a diffusion-limited process is present at the surface. This is likely caused bya combination of diffusion-limited charge transfer, reaction limited charge transfer, and ionicadsorption is present at the surface of the gold electrodes. These double-layer charging effectscan be modeled using fractional order operators [17]. For all experimental measurements, themembrane contained negligible defects. Fig. 4.16 presents an example of the experimentallymeasured and numerically predicted impedance for the tethered membrane. As seen, the pre-dicted impedance is in excellent agreement with the experimental measured impedance andis consistent with a membrane containing negligible defects. Insights into how changes in theparameters Gm, Cm, Cdl, and Re effect the impedance is provided in Sec. 3.4.4.10−1 100 101 102 103304050607080906 Z(f)[o ]f [Hz] 10−1 100 101 102 103104105106107Z(f)[Ω]f [Hz]Figure 3.7: The measured and predicted impedance (phase is represented by ∠Z(f) in degreesand magnitude by Z(f)) of the 10% tether density DphPC bilayer membrane. The solid line isthe predicted and the dotted the experimentally measured. The grey and black colours indicatethe impedance for two identically constructed membranes. All predictions are computed using(3.1). The experimental results are extracted from [3].To test which reaction mechanism (3.2) or (3.3) produces results in agreement with theexperimentally measured data, the mean-absolute percentage error (MAPE) for each of thereaction mechanisms is computed for varying concentrations of PGLa, and for charged anduncharged membranes. For reaction mechanism (3.2) the MAPE is 3.7%, and for (3.3) theMAPE is 7.2%. This suggests that for the analyte concentrations up to 40 µM the PGLapeptide binds to the membrane surface via the monomer binding mechanism illustrated inFig. 3.4. This is expected as the dimer binding mechanism (Fig. 3.4) is suggested for peptide tolipid ratios above 1:200. Given the area per lipid in the PFMP is approximately 0.68 nm2 [3],there are approximately 3×1012 lipids in the PFMP. For the dimer binding mechanism to occurthe surface concentration is expected to be larger then 12 nmol/m2 which is never reached.In Fig. 3.8 the experimentally measured and numerically predicted conductance for vary-ing concentrations of PGLa is presented. Initially the conductance increases as a result ofPGLa peptides diffusing to the membrane surface, binding via the monomer binding mecha-nism (Fig. 3.4), and then translating to the transmembrane configuration (Fig. 3.5) increasingthe membrane conductance. As shown in Fig. 3.8, at about 500 s the conductance of the473.4. Numerical and Experimental Resultsmembrane begins to decrease. This suggests that PGLa pores begin to close and prevent theformation as new PGLa pores. Since PGLa has a positive charge of +4 as a result of the Lysineresidues, as the membrane becomes saturated with PGLa this causes the overall charge of themembrane to decrease inhibiting the insertion of PGLa into the membrane. The estimatedPGLa pore conductance Gp is in the range of 0.6 to 3.5 pS, in agreement with the expectedpore conductance from [129, 130]. As expected the association coefficient ka (3.2) is four ordersof magnitude larger than the protomer formation rate constant kp. This is expected as thebinding of the PGLa to the surface (Fig. 3.4) does not require the translocation of the peptideto the transmembrane state. As suggested from the results in Fig. 3.5, the translocation likelyinvolves the peptide diffusing into a transient aqueous pore which is a slower process then thepeptide directly binding to the surface. The rate of closing kc (3.2) is large suggesting thatPGLa pores only form transiently in the uncharged membrane. This provides an explanationfor why the transmembrane state of the PGLa has not been observed at physiological temper-atures using NMR techniques [116]. The estimated solution diffusion coefficient of PGLa (Dain (3.4)) is in the range of 2 nm2/ns to 5 nm2/ns.0 1,000 2,000 3,000 4,000 5,0000246Time [s]ConductanceG m(t)[µS] 40 µM PGLa30 µM PGLa20 µM PGLa10 µM PGLaFigure 3.8: Experimentally measured and numerically predicted conductance for DphPC teth-ered membrane with 10, 20, 30, and 40 µM of PGLa. The predictions are made using (3.4), (3.5)with the reaction mechanism given by (3.2) and simulation parameters provided in Table 3.4.The experimental results are extracted from [4].To gain insight into the effect the membrane charge has on the dynamics of PGLa poreformation, a membranes with varying concentration of charged POPG lipids are constructed.Fig. 3.9 presents the response of the 10% tethered membrane with charged (POPG lipids) anduncharged (DphPC) lipids resulting from the addition of 30 µM PGLa. As seen the numericallyestimated results are in excellent agreement with the experimentally measured conductance.Just as with the uncharged membrane, the peptides diffuse to the membrane surface, bind viathe monomer binding mechanism (Fig. 3.4), and then enter the transmembrane configuration(Fig. 3.5) increasing the membrane conductance. As expected, the negatively charged POPGlipids promote the binding of positively charged PGLa peptides–as % of POPG lipids increase483.4. Numerical and Experimental Resultsthe membrane conductance increases. In comparing Fig. 3.8 with Fig. 3.9, it is clear thatPGLa has an affinity for forming pores in biological membranes containing negatively chargedlipids typically found in bacterial cells. Using the reaction rate model we find that the rateof protomer formation k1 in (3.2) increases as the negative charge of the membrane increases.However, the rate of pore closure (kc in (3.2)) decreases as the negative charge of the membraneincreases. Therefore we conclude that as the negative charge of the membrane increases thereis an increase in the number of PGLa pores and pore lifetime. This makes PGLa especiallyeffective for killing biological membranes containing a net negative charge.0 1,000 2,000 3,000 4,000 5,0000102030Time [s]ConductanceG m(t)[µS] 50% POPG40% POPG30% POPG20% POPGFigure 3.9: Experimentally measured (dotted) and numerically predicted (solid line) membraneconductance for tethered membranes composed of 10%, 20%, 30%, 40%, and 50% POPG witha PGLa concentration of 30 µM. The predictions are made using (3.4), (3.5) with the reactionmechanism given by (3.2) and simulation parameters provided in Table 3.5. The experimentalresults are extracted from [4].3.4.4 Experimental Setup, Model Validation, and Model ParametersIn this section the experimental setup of the PFMP and numerical methods utilized to producethe results presented in Fig. 3.4 to Fig. 3.9 are provided. The validation of the PFMP andexperimental measurements to estimate the reaction mechanism of α-Hemolysin is also providedin this section.Experimental MeasurementsThe experimental result are reported in [4]. For completeness, we summarize the main parame-ters. All experimental measurements were performed at a temperature of 20◦ C in a phosphatebuffered solution with a pH of 7.2 and a saline solution of 0.15 M composed of Na+, K+, andCl−. A pH of 7.2 was selected to match that typically found in the cytosol of biological cells.The formation of the tethered bilayer lipid membrane can be found in [3, 16]. The negativelycharged palmitoyl-oleoyl-phosphatidylglycerol (POPG) lipid membrane was constructed using493.4. Numerical and Experimental Resultsan ethanolic solution containing archaebacterial lipids (70% zwitterionic C20 diphytanyl-ether-glycero-phosphatidylcholine lipid and 30% C20 diphytanyl-diglyceride ether) mixed with 0-50%POPG lipids. The impedance of the PFMP is recorded using a tethaPod (SDx Tethered Mem-branes) which measures the impedance using a 20 mV excitation at frequencies of [1000, 500,200, 100, 40, 20, 10, 5, 2, 1, 0.5, 0.2,0.1] Hz.The protein toxin α-Hemolysin was obtained from Sigma-Aldrich (St. Louis, MO). Theantimicrobial peptide PGLa was synthesized using solid-phase Fmoc protocols [147, 148] on anApplied Biosystems (Carlsbard, CA) 433A instrument using a reverse-phase HPLC [149].Pore Formation Dynamics of α-HemolysinTo validate the PFMP and model we measure the pore formation dynamics of α-Hemolysin. InFig. 3.10 the experimentally measured and numerically predicted conductance for α-Hemolysinis presented. As seen the experimental data is in excellent agreement with the numericallyestimated conductance. The reaction mechanism for α-Hemolysin [128] is given by:ak1a−⇀↽−kdm1kp−→ p1 p1 + pi k1−→ pi+1 mpn kc−→ cfor i = 1, 2, . . . , n− 1 (3.9)with the parameters defined below (3.2). The association rate constant k1a in (3.9) is given byk1a = ka(mmax−m1−∑i pi−nmc) which accounts for the intermediate species pj leading to poreformation. For α-Hemolysin the number of protomers is known and is given by n = 7 [120, 121].As expected the association rate constant ka is approximately three orders of magnitude largerthan the protomer formation rate constant kp. The rate limiting step is the diffusion of theα-Hemolysin monomers on the surface of the membrane. Since the conductance of α-Hemolysinpores is known to be 92±12 pS [121], we are able to estimate the population of pores from theconductance change. There is approximately 175±25 thousand pores present in the membraneafter the diffusive effects become negligible for t > 1500 s.Impedance Sensitivity to Variations in Model ParametersIn this section insight is provided on how variations in the parameters Cm, Cdl, Re and Gm in(3.1) can be detect from the experimentally measured impedance. Fig. 3.11 provides the nu-merically computed impedance for variations in Cm, Cdl, and Re. As seen from Fig. 3.11(a)-(c),using an excitation frequency from 0.1 Hz to 1 kHz allows the estimation of the parametersCm, Cdl, and Re. In Fig. 3.11(a) we see that the major effect of Cm on the impedance oc-curs at frequencies above 1 Hz, and from Fig. 3.11(b) the effects of Cdl are pronounced forfrequencies below 10 Hz. This results as Cm is about a factor of 10 smaller then Cdl for thetethered membrane platform. Fig. 3.11(c) shows that Re only causes measurable effects on the503.4. Numerical and Experimental Results0 1,000 2,000 3,000 4,000 5,000 6,000051015Time [s]ConductanceG m(t)[µS]3 µM α-HemolysinFigure 3.10: Experimentally measured (dotted) and numerically predicted (solid line) conduc-tance for DphPC tethered membrane and α-Hemolysin concentration of 3 µM. The predictionsare made using (3.4), (3.5) with the reaction mechanism given by (3.9) and simulation param-eters provided in Table 3.3. The experimental results are extracted from [4].phase at frequencies above 110 Hz. Note that once the parameters Cm, Cdl, and Re have beenestimated they do not change during the experimental measurements as such changes wouldindicate a catastrophic change in the tethered membrane stability. The only parameter thatvaries throughout the experimental measurements is the membrane conductance Gm. Fig. 3.12illustrates how changes in Gm can be estimated from the experimentally measured impedanceresponse. As seen, using an excitation potential of 0.1 Hz to 1kHz allows the measurement ofGm in the range of 0.1 µS to 100 µS.Coarse-Grained Molecular Dynamics: Method and ParametersAll CGMD simulations were implemented in GROMACS version 4.6.2 [150–152]. For all pro-duction runs the Berensden temperature coupling is used with a temperature of 323 K, anda time constant of 0.3 ps. Note that the temperature coupling was also implemented usingthe velocity rescaling algorithm [153] with a time constant of 0.5 ps with the results in agree-ment with the results using the Berensden thermostat. The Berendsen semi-isotropic pressurecoupling is used with a time constant of 3.0 ps, compressibility of 3×10−5 1/bar, and a refer-ence pressure of 1.0 bar [154]. The timestep of the simulation is 20 fs with the electrostaticinteractions smoothly shifted from zero at 12 A˚ and Lennard-Jones interaction from 9-12 A˚.The membrane is modelled using 512 dipalmitoylphosphatidylcholine (DPPC) molecules. Notethat DPPC has an identical structure to DphPC, and a similar structure to GDPE when usingthe CGMD representation as fine details such as the phytanyl tails in the DphPC and GDPEare equivalent to the palmitoyl tails in DPPC. The 512 lipid CGMD DphPC membrane isconstructed by replicating the equilibrated 128 DPPC bilayer, from the MARTINI website1,twice in the X and Y directions. The 512 DphPC membrane is solvated using CGMD waterbeads and energy minimized followed by an equilibration in NPT for 200 ns to produce theequilibrated membrane structure. The dimensions of the simulation cell containing the mem-1http://md.chem.rug.nl/cgmartini/index.php/downloads513.4. Numerical and Experimental Resultsbrane are 126A˚×129A˚×150A˚ corresponding to X×Y×Z coordinate axis. The solvent solutionsurrounding the peptide and membrane surface is composed of water molecules and Na+ andCl− ions to make the solvent a 0.1 M NaCl solution and also to neutralize the charge on thepeptides.Surface Binding of PGLa: To study the binding mechanism of PGLa, two simulationsare constructed for the monomeric binding and dimer binding illustrated in Fig. 3.4. For themonomer binding a single PGLa peptide is placed 17 A˚ above the surface of the membrane. Forthe dimer binding two PGLa peptides where placed 17 A˚ above the surface of the membranewith a center of mass spacing between each peptide of 28 A˚. After energy minimization, theproduction run was carried out for a simulation time horizon of 1 µs.Transmembrane Insertion of PGLa: To construct the transient aqueous pores in Fig. 3.5which allow the PGLa to translocate from the membrane surface to the transmembrane state,we employ the method outlined in [155] to construct the aqueous pores. The Berendsen semi-isotropic pressure coupling is used with a time constant of 3.0 ps, compressibility of 3×10−51/bar, and a reference pressure of 1.0 bar in the direction normal to the membrane surface.The lateral pressure is held at −50 bar until a transient pore has formed. The negative pressurepromotes the formation of transient aqueous pores in the membrane. After a transient porehas formed and the PGLa have diffused into the aquous pore, the lateral pressure is set to 1.0bar allowing the transient aqueous pore to close. The production run for pore closure has asimulation time horizon of 500 ns.Oligomerization of Transmembrane PGLa: To study the oligomerization process of PGLawe place 4 PGLa molecules in the transmembrane state, as illustrated in Fig. 3.6, using themethod outlined in [136]. PyMOL (The PyMOL Molecular Graphics System, Version 1.3Schro¨dinger, LLC) is used to place the peptides in the transmembrane state and in the interiorof the membrane. The system is equilibrated in the NPT ensemble for 20 ns. After energyminimization the production run is carried out for 250 ns.Diffusion: To estimate the diffusion of water, lipids, and PGLa complexes we compute theensemble-averaged time-dependent mean square displacement (MSD) 〈(x−xo)2〉 with xo denot-ing the initial position, x the position of the complex t later, and ensemble average taken overtime. The MSD is computed with respect to the center-of-mass of the molecule. The MSD isrelated to the diffusion coefficient by 〈(x−xo)2〉 = 4D(t−to)α where we allow for the possibilityof subdiffusion (i.e. α < 1) as compared with normal diffusion α = 1. To reduce the noise ofthe estimated diffusion D, we average the MSD of sub-trajectories of t, as done in [156]. Thediffusion coefficient are computed from production runs of 250 ns. For all diffusion coefficientsα ≈ 1 indicating the peptide satisfies standard Fickian diffusion (i.e. normal diffusion).523.4. Numerical and Experimental ResultsContinuum Model: Numerical Method and ParametersThe governing equations (3.4) and (3.5) with boundary conditions (3.6) and initial conditions(3.7), are solved numerically with the commercially available finite element solver COMSOL4.3a (Comsol Multiphysics, Burlington, MA). The simulation domain is meshed with approx-imately 28,199 triangular elements constructed using an advancing front meshing algorithm.(3.4) and (3.5) are numerically solved using the multifrontal massively parallel sparse directsolver [107] with a variable-order variable-step-size backward differential formula [108]. (3.8)is used to compute the pore conductance with the integration done in the region ∂Ωsurf. Thecomputational domain of the continuum model is provided in Fig. 3.13. A two-dimensionalsimulation domain is considered as the chamber width of the PFMP is W = 3 mm and cham-ber height he = 0.1 mm. As shown in [157], for he/W < 0.1 the variation in concentrationalong the width of the chamber is negligible. For the PFMP the aspect ratio is he/W = 0.03,therefore a two-dimensional domain can be used to model the reaction-diffusion dynamics inthe PFMP.Table 3.2: Model Parameter for PFMP.Symbol Definition Valuehc Inlet chamber height 4 mmLin Inlet chamber length 2 mmL Channel buffer length 1 mmhe Channel height 0.1 mmLe Electrode length 0.7 mmLout Outlet chamber length 20 mmhout Outlet chamber height 4 mmThe maximum concentration of PGLa in solution is computed by multiplying the molecularweight of PGLa (1970 g/mol) by the average protein specific volume of 0.73 cm3/g [158]. Themaximum concentration of PGLa in solution is 695 mM–this corresponds to ra = 1.33 nm in(3.4). The maximum surface concentration of membrane bound PGLa and protomers is takenas corresponding to 1% of the total molar concentration of the tethered membrane lipids. Eachlipid in the tethered membrane has a surface area of 0.68 nm2, therefore for a 2.1 mm2 membranethere are approximately 3 × 1012 lipids in the surface layer. Therefore the maximum surfaceconcentration of membrane bound PGLa and protomers is 2.5×10−8 mol/m2. This correspondsto an effective surface radius for each membrane bound PGLa and protomer (i.e. rm and ri in(3.5) to be 8 nm. The maximum surface concentration for PGLa mmax, defined below (2.1),was selected to match that of the maximum surface concentration of Cytolysin A [126]. Thereaction-diffusion parameters for numerical results presented in Fig. 3.10, Fig. 3.8, and Fig. 3.9are provided in Table 3.3, Table 3.4, and Table 3.5 respectively.533.4. Numerical and Experimental ResultsTable 3.3: Model Parameter for α-Hemolysin (Fig. 3.10)Symbol Definition Valuemmax Saturated surface concentration 5× 10−10 mol/m2ka Adsorption rate constant 1× 108 m2/smolkd Desorption rate constant 1 1/skp Rate of protomer formation 1× 105 1/sk1 Rate of protomer binding 1.96× 104 m2/smolkc Rate of closing 0 1/sDa Analyte diffusion coefficient 1.2 nm2/nsTable 3.4: Model Parameter for Varying PGLa (Fig. 3.8)Symbol Definition Valuera Effective radius of PGLa 1.33 nmmmax Saturated surface concentration 5× 10−10ka Adsorption rate constant 5000 m2/smolkd Desorption rate constant 0 1/skd Desorption rate constant 0 1/skp Rate of protomer formation 0.5 1/sk1 Rate of protomer binding 0.5 1/sn n in (3.2) 1Da Analyte diffusion coefficient 2-5 nm2/nsPGLa concentration ao: 10 µM 20 µM 30 µM 40 µMm m in (3.2) 2 2 3 3kc Rate of closing 1× 102 m2/smol 1.2× 106 m2/smol 26× 1015 m4/smol2 1.1× 1015 m4/smol2Table 3.5: Model Parameter for Varying POPG (Fig. 3.9)Symbol Definition Valuera Effective radius of PGLa 1.33 nmmmax Saturated surface concentration 5× 10−10ka Adsorption rate constant 1× 105-2× 105 m2/smolkd Desorption rate constant 0 1/skp Rate of protomer formation 10 1/sn n in (3.2) 3k1 Rate of protomer binding 0.6× 1015-1.5× 1015 m4/smol2m m in (3.2) 1kc Rate of closing 1× 10−3- 1.3× 10−3 1/sDa Analyte diffusion coefficient 3 nm2/ns543.4. Numerical and Experimental Resultsf [Hz]10-11001011021036Z(f)[o]30405060708090f [Hz]10-1100101102103Z(f)[+]104105106107100Ω1500Ωf [Hz]10-11001011021036Z(f)[o]30405060708090f [Hz]10-1100101102103Z(f)[+]104105106107108190 nF100 nFf [Hz]10-11001011021036Z(f)[o]20406080100f [Hz]10-1100101102103Z(f)[+]1041051061075 nF20 nFCmCdlReabcFigure 3.11: Panel (a) illustrates how the impedance changes for the Cm values of 5 nF, 10 nF,15 nF, and 20 nF with all other parameters fixed. (b) illustrates how the impedance changesfor the Cdl values of 100 nF, 130 nF, 160 nF, and 190 nF with all other parameters fixed. (c)illustrates how the impedance changes for the Re values of 100 Ω, 500 Ω, 1000 Ω, and 1500 Ωwith all other parameters fixed. The predicted impedance (phase is represented by ∠Z(f) indegrees and magnitude by Z(f)) is computed using (3.1).f [Hz]10-11001011021036Z(f)[o]30405060708090f [Hz]10-1100101102103Z(f)[+]1021041061080.1µS 1µS10µS 100µSFigure 3.12: The predicted impedance (phase is represented by ∠Z(f) in degrees and magnitudeby Z(f)) for varying membrane conductance Gm with Cm, Cdl, and Re fixed. All predictionsare computed using (3.1).553.4. Numerical and Experimental ResultsLeL LLin Louthehc∂Ωsurf∂Ωb ∂Ωb∂ΩinFigure 3.13: PFMP and a schematic of the computational domain for the generalized reaction-diffusion continuum model. The parameters are defined in Table 3.2. ∂Ωb is the boundary of thetethered membrane illustrated by the black boxes, and ∂Ωin is the analyte input flow-chamberindicted in gray.563.5. Closing Remarks3.5 Closing RemarksIn this chapter a possible reaction mechanisms for the pore formation dynamics of PGLa incharged and uncharged tethered membranes that mimic biological membranes is provided. Aswas shown, PGLa targets and increases the permeability of biological membranes containinga net negative charge. We show that PGLa not only increases the number of pores in nega-tively charged membranes, but the lifetime of conducting pores also increases compared to thelifetime of PGLa pores in uncharged membranes. Using coarse-grained molecular dynamics weshow that PGLa monomers and dimers can bind to the membrane surface, transition to thetransmembrane conformation via transient aqueous pores, and oligomerize once in the trans-membrane conformation. These results were justified using experimental measurements from atethered membrane in combination with a three level atomistic-to-observable model. The modelconsists of coarse-grained molecular dynamics, a continuum model composed of a generalizedversion of Fick’s law of diffusion coupled with surface reaction-diffusion equations, and a frac-tional order macroscopic model. The model was validated using experimental measurementsof the pore forming toxin α-Hemolysin. Though we apply the pore formation measurementplatform and atomistic-to-observable model to the study of pore formation dynamics of PGLa,the platform and modeling methodology are general and can be used for other pore formingpeptides and proteins of interest.574Controlled ElectroporationMeasurement Device4.1 IntroductionThis chapter focuses on the construction, operation, and predictive modeling of a cell-basedtethered membrane device that provides a precisely controllable platform for electroporationstudies. The device consists of membranes engineered from synthetic archaebacterial lipids, andlipids extracted from prokaryotic and eukaryotic cells. The tethers mimic cytoskeletal supportsin biological membranes thereby facilitating in vivo measurements of electroporation. Alsoin this chapter a novel atomistic-to-observable predictive model comprising of three levels ofabstraction: coarse-grained molecular dynamics (CGMD) that incorporates membrane tethers,a Generalized Poisson-Nernst-Planck continuum model, and a fractional order macroscopicmodel is proposed. Experimental measurements using this in vivo device together with thepredictive model yield important insights into the effect archaebacterial, Escherichia coli , andSaccharomyces cerevisiae lipids, the electrolyte concentration, and the bioelectronic interfacehave on electroporation.4.1.1 Measurement of ElectroporationElectroporation is the phenomenon where aqueous pores form spontaneously in a cell mem-brane when a high transmembrane potential is applied. Electroporation facilitates the passageof otherwise impermeable molecules across the membrane into a cell and is used to catalysethe uptake of chemotherapeutic agents, DNA molecules, and neuron specific proteins in drugdelivery applications. It is reported [19] that electroporation can circumvent poor delivery ofmedications to the central nervous system and hence a potential tool in treating Alzheimers,Parkinson’s disease, and some brain cancers. Though widely used, the electroporation processremains poorly understood. This is hindering the development of novel electrochemotherapyprotocols. Key to the design of successful electrochemotherapy protocols is the constructionof a robust and controllable platform for the study of electroporation, and an atomistic-to-observable model of the electroporation process.584.1. IntroductionTwo classes of platforms exist for the study of electroporation. The first is in vivo cells [39,40, 43, 44] which provide a physiological system for validation. A complication with using cellsfor model validation is that it is impossible to fully define the physiological environment whichaffects properties associated with electroporation. The second class is synthetic bilayer lipidmembranes [3, 41, 42] which model the physiological response of real cell membranes. Syntheticmembranes benefit as the physiological environment is controllable, however components of realcell membranes such as the cytoskeletal network, and membrane species including proteins andpeptides are not included. This motivates the search for a well defined model to study electro-poration that incorporates the components of natural cell membranes in a highly controllablephysiological environment.Models for the complex dynamics of electroporation at the atomistic level are typicallyconstructed using molecular dynamics. Recent results using molecular dynamics have shownthat the mechanism of pore formation in symmetric and asymmetric membranes is dependenton the lipid composition [159–161]. That is, pure DphPC membranes have a higher resistanceto electroporation compared to that of a DPPC membrane as a result of the mobility of thehydrophobic tails of the DphPC lipid [160]. Using molecular dynamics it is shown that theEscherichia coli (E. coli) membrane has a higher resistance to electroporation compared to thatof Staphylococcus aureus membranes as a result of the presence of lipopolysaccharides [161]. Thisresult suggests that membranes containing lipopolysaccharides would have a higher resistanceto electroporation compared to prokaryotic membranes that do not contain lipopolysaccharides.To our knowledge no experimental evidence has been provided to support these results. Alsomolecular dynamics has not been applied to the study of electroporation in the presence ofcytoskeletal supports, even though a cytoskeleton network is adjacent to all cell membranes.In such cases, standard molecular dynamics is computationally intractable and obtaining ameaningful structure to function relation requires use of CGMD. Recent models for the dynamicsof electroporation at the continuum and macroscopic levels have employed the GeneralizedPoisson-Nernst-Planck (GPNP) and asymptotic approximations to the Smoluchowski-Einsteinfor electroporation [3]. The macroscopic and continuum model can account for the dynamicsof asymmetric electrolytes, multiple ionic species, Stern and diffuse layers present. Howevermacroscopic and continuum models are not suitable for computing molecular level parameterssuch as diffusion.4.1.2 Main Results and OrganizationA significant contribution of this chapter is to devise an atomistic-to-observable model thatlinks the results from molecular dynamics with the observed macroscopic current response ofthe CED. Such an atomistic-to-observable model for the controlled electroporation measurementdevice (CED) gives valuable insight into understanding and controlling electroporation. The594.1. Introductionformation and operation of the CED is provided in Sec. 4.2. The CED considered in this chapteris composed of engineered archaebacterial lipids, and lipids from E. coli and Saccharomycescerevisiae (S. cerevisiae) membranes. The number of tethers, physiological environment, andmembrane composition are all controllable in the CED. The atomistic-to-observable modellinks the atomistic dynamics to experimental measurements from the CED. The model weconsider is composed of three levels of abstraction: atomistic, continuum, and macroscopic.The atomistic dynamics of the CED are modeled using coarse-grained molecular dynamics(CGMD) [74, 75]. The continuum model is composed of the GPNP system of equations whichare used to link the results from the CGMD to the macroscopic model. The macroscopicmodel is composed of a system of fractional order differential equations that are dependent onthe computed parameters from CGMD and GPNP models and links the simulation results withexperimental measurements. The dynamic model of the CED is provided in Sec. 4.3. In Sec. 4.4several key findings related to the electroporation and bioelectronic interface are presented usingexperimental measurements from the CED and the three level atomistic-to-observable model(Fig. 4.1). These include:1. In contrast to previous molecular dynamics results [20], no anomalous diffusion was detectedat the surface of the membrane or in the tethering reservoir between the gold and membranesurfaces. This allows the transport of ions away from the electrode surface in the CED to bemodeled using continuum theories that do not include fractional order operators [17]. Forexample the GPNP model can be used to model the electrodiffusive dynamics of ions in thetethering reservoir.2. Membrane thickness, lipid diffusion adjacent to the tethering reservoir, pore density, thefree energy barrier for lipid flip-flop, and effective dielectric permittivity of the membraneare dependent on the tether density. This suggests that when performing electroporationtherapies the tethering density (i.e. density of cytoskeletal supports) is an essential criterion.3. The conductance of aqueous pores in the CED is linearly proportional to the radius of thepore and not the radius squared as typically assumed [21]. This unique feature is a result ofthe proximity of the membrane to the gold surface.4. Experimental measurements show that a combination of diffusion-limited charge transfer,reaction limited charge transfer, and ionic adsorption is present at the surface of the goldelectrodes. These double-layer charging effects can be modeled using fractional order oper-ators [17].5. The predictions that we obtain using the atomistic-to-observable model accurately matchactual experimental data for CEDs from DphPC, E. coli, and S. cerevisae lipids for differentexcitation potentials and tethering densities. This supports the use of molecular dynamicsto go from structure to function.604.2. Controlled Electroporation Measurement Device (CED): Formation and OperationThe above results provide valuable insights into precise control of electroporation, and into thediffusive transport of lipids and ions in natural membranes. They also yield design rules forfuture membrane based biosensors and novel electroporation therapies. Closing remarks areprovided in Sec. 4.54.2 Controlled Electroporation Measurement Device (CED):Formation and OperationThe CED is constructed using the rapid solvent-exchange technique developed by co-authorCornell [10] presented in Sec. 3.2. A schematic of the constructed CED is provided in Fig. 4.1.The density of tethers, physiological environment, and membrane composition are all adjustablein the CED. Using the fundamentally different physical formation process of the CED comparedto that of black lipid membranes (BLMs) allows the CED to have a lifetime of several monthsand the ability to withstand excitation potentials of up to 800 mV. Typically, BLMs have alifetime of tens of minutes. The CED is composed of engineered archaebacterial lipids, tethers,spacers, and lipids from real cell membranes such as E. coli and S. cerevisiae. We denote thetethering density as the ratio of spacer molecules to tethering molecules. For example, for a10% tethered membrane, for every 9 spacer molecules there is 1 tether molecule. The long lifeof the CED is a result of how the tethers and spacers are bonded to the inert gold surface. Thetethers and spacers both contain a benzyl disulphide attachment chemistry to the gold. Thesulphur bonded to the benzyl, bonds to the gold surface with the disulphide bond maintained.This bonding structure has been detected experimentally from X-ray photoelectron spectra.The use of the disulphide has two advantages: the thiols do not oxidize on storage allowing themembrane to have a lifetime of several months, and maintains the spacing between the spacersand tethers [4, 9–12]. Measurements with the CED are performed by applying a drive voltageVs between the gold electrodes and measuring the current response I(t).614.3. Dynamic ModelFigure 4.1: Schematic of the controlled electroporation measurement device and atomistic-to-observable model. The “Electronic” block applies a potential between the electrodes, thisincreases the transmembrane potential Vm, and records the current response I(t). Gp is theconductance of the aqueous pore. In the atomistic-to-observable model the coarse-grainedmolecular dynamics (CGMD) is used to compute the diffusion D, thickness of the membranehm, surface tension σ, and line tension γ. The Continuum model is used to compute the poreconductance Gp and electrical energy required to form a pore Wes. The Macroscopic model isused to relate Gp,Wes, σ, and γ to the experimentally measured current I(t). In the CGMDpanel, the yellow beads model the bioelectronic interface, the translucent blue beads the water,and the green beads the tethered membrane.4.3 Dynamic ModelIn this section the atomistic-to-observable model fo the CED is provided. The atomistic-to-observable model is designed to link the results from atomistic simulations to the resultsobtained from experimental measurements. The model is composed of coarse-grained moleculardynamics (CGMD) [74, 75], a Generalized Poisson-Nernst-Planck (GPNP) continuum model,and a fractional order macroscopic model as illustrated in Fig. 4.1. The CGMD model isconstructed using the MARTINI force-field [74, 75] which combines the speed-up benefits of a624.3. Dynamic Modelsimplified model with the resolution obtained by atomistically detailed models such as moleculardynamics. Using the CGMD model allows the diffusion tensor D, thickness of the membrane hm,surface tension σ, number of pores N , and line tension γ to be computed. To link the CGMDresults with the macroscopic model, the aqueous pore conductance Gp and electrical energyrequired to form a pore Wes are computed using the GPNP continuum model. The fractionalorder macroscopic model is used to link the GPNP continuum model with the experimentallymeasured current response of the CED.4.3.1 Coarse-Grained Molecular Dynamics ModelHere the CGMD model of the CED is provided which is used to compute important electropo-ration parameters such as anisotropic diffusion, surface tension, line tension, and pore density.CGMD System SetupTo construct the CGMD model we map the molecular components of the CED into the MAR-TINI [74, 75] force field. The molecular components include the zwittrionic C20 diphytanyl-ether-glycero-phosphatidylcholine lipid (DphPC), C20 diphytanyl-diglyceride ether lipid (GDPE),benzyl disulphide connected to an eight-oxygen-ethylene-glycol group terminated by a C20 hy-drophobic phytanyl chain (tether), and benzyl disulphide connected to a four-oxygen-ethylene-glycol group terminated by an OH (spacer), and the gold surface. The proximal layer of theCED is composed of tethered and a 30% GDPE to 70% DphPC ratio of lipids. In our CGMDmodel for the DphPC tethered membrane the distal layer is composed of 30% GDPE and 70%DphPC. To gain insight into the effect of tethers we constructed a 0% and 25% tether density(i.e. a 25% tether density defines that for every 3 spacer molecules, there is 1 tether molecule)CGMD model. How each is mapped to the CGMD force-field is described below. Note thatunless otherwise specified, the bead types and interactions are provided in [74, 75].Lipids: The phosphatidylcholine headgroup of the DphPC lipid is represented by two beads:the positive choline by the Qo bead, and the negative phosphate by the Qa bead. The ether-glycol is represented by a SNa bead, and each of the phytanyl tails by four C1 beads.Thephytanyl and ether glycerol moieties of GDPE are represented by the same mapping as for theDphPC, however the hydroxyl headgroup of GDPE is represented by a P4 bead. In total theDphPC lipid is composed of 12 beads, and the GDPE lipid by 11 beads.Tethers and Spacers: The eight ethylene glycol COC molecules of the tethers are representedby 8 PEG beads. The interaction of the PEG beads is provided in [77, 162]. The benzyldisulphide group is represented by a C5 bead which has the highest polar affinity of the standardMARTINI CG beads [75]. The phytanyl tail of the tethers is mapped using an identical methodas for the GDPE and DphPC lipids. The spacers are mapped using an identical method as thetethers however the hydroxyl group is represented by the P4 bead.634.3. Dynamic ModelGold Surface: The gold surface is composed of a square lattice with custom Pf beads. Thedistance between adjacent beads is 0.3 nm. The interaction of the Pf bead is designed toreduce the effects of excess adsorption to the surface. The interaction between Pf and P4 is 1/3the value between P4 and P4, and the interaction between Pf and other bead types is ∼ 12%of the MARTINI value between P4 and respective bead types. The following interactions areexcluded: interaction between Pf beads, and between the C5 beads of the tethers and spacers,and P4 and Qo beads of the lipids. Note that a similar interaction is used in [78] to representthe gold surface in the MARTINI force-field.CGMD and Simulation MethodThe molecular dynamics simulations were performed using GROMACS [163] version 4.6.2 (dou-ble precision) with the MARTINI force field [74, 75]. Unless otherwise stated, the CGMD sim-ulation parameters are provided in [77, 78]. The interaction of the CGMD beads are definedby the Lennard-Jones (LJ) potential, and harmonic potentials are utilized for bond and angleinteractions. A shift function is added to the Coulombic force to smoothly and continuouslydecay to zero from 0 nm to 1.2 nm. The LJ interactions were treated likewise except that theshift function was turned on between 0.9 nm and 1.2 nm. The grid-type neighbour searchingalgorithm is utilized for the simulation–that is, atoms in the neighbouring grid were updatedevery ten time steps. The equations of motion are integrated using the leapfrog algorithm witha timestep of 20 fs. Periodic boundary conditions are implemented in three-dimensions. Sim-ulations are performed in the NVT ensemble using a temperature of 320 K. The temperatureis held constant using a velocity rescaling algorithm [153] with a time constant of 0.5 ps. Thelipids, tethers, spacers and water molecules are coupled separately for temperature control.All the systems studied here were first energy minimized using the steepest descent method inGROMACS. A 50 ns equilibration run is performed prior to the production run. Productionruns are performed for a simulation time of 1.5 µs. Visualization of the results are reportedusing VMD and PyMOL.Line Tension and Surface TensionTo compute the line tension γ of the membrane we use the procedure provided in [164]. Theline tension can be computed from the ribbon like structure (Fig. 4.2) usingγ = 12〈LxLy[Pxx + Pyy2 − Pzz]〉(4.1)with Pxx, Pyy, Pzz the diagonal elements of the pressure tensor, Lx and Ly the simulation sizein the x and y directions respectively, and 〈· · · 〉 denoting the ensemble average over time.To construct the lipid structure in (Fig. 4.2) an intact bilayer containing 320 lipids, in a644.3. Dynamic Model70% DphPC and 30% GDPE composition is used. The hydrophilic interior of the bilayer isinitially adjacent to the x and z dimensions of the simulation cell. The simulation cell is thenexpanded in the x direction from 14 nm to 16 nm, and in the y direction from 10 nm to 13 nmto ensure the membrane forms an edge. Initially a 50 ns equilibration run was performed toallow the edge to form, this was followed by a 250 ns production from from which γ (4.1) canbe estimated. Simulation are performed in a NPxyLzT ensemble at a temperature of 320 K.Temperature is kept constant using the velocity rescaling algorithm [153] with a time constantof 0.5 ps. Pressure is coupled semi-isotropically using the weak coupling scheme [153] with atime constant of 3 ps, compressibility of 0.3 nm2/nN, and a reference pressure of 100 kN/m2.Figure 4.2: Ribbon structure of 0% tethered DphPC membrane. Lipid tails are represented bythe green beads, NC3 bead is displayed in blue, the PO4 bead in orange, OH bead in red, theCOC bead as pink, and the water beads as a translucent blue. The coloring scheme of the axisis red for x, blue for y, and green for z. Note that this axis is only used for computing the linetension of of the membrane as discussed in Sec. 4.3.1.The surface tension of the membrane is computed using [164]:σ = 12〈Lz[Pzz −Pxx + Pyy2]〉(4.2)with the parameters defined below (4.1). The evaluation of (4.2) is performed in the NAPxyTensemble using a total production run of 250 ns.Pore Density and Membrane StabilityWe have used the umbrella sampling method [165] to calculate the free energy profile for movingthe phosphate of a single DphPC lipid into the center of different mixed bilayers of varying654.3. Dynamic Modeltethering densities. The umbrella potential acts on the center of mass of the phosphate groupof the DphPC lipid with a harmonic restraint with a force constant of 500 kJ/mol/nm2 in thedirection normal to the bilayer plane. There were 0.15 nm and 0.1 nm spacing between biasingpotentials, resulting in 37 and 48 parallel simulations for 0% and 25% tethered membranes,respectively. The starting structures corresponding to each umbrella window were created bypulling a lipid to their respective position using an umbrella potential with a force constantof 500 kJ/mol/nm2 in a 1 ns simulation. Each umbrella window was then equilibrated for 50ns with a 1000 kJ/mol/nm2 force constant, followed by a 1 µ-sec simulation. The potential ofmean force (PMF) profile was constructed from the biased distributions of the center of massof the lipids using the weighted histogram analysis method [166] with a relative tolerance of10−4.4.3.2 Generalized Poisson-Nernst-Planck Continuum ModelThe GPNP mesoscopic model utilizes the results from the CGMD model to evaluate importantparameters for the macroscopic model such as pore conductance Gp and electrical energy re-quired to form a pore Wes, refer to Fig. 4.1. Gp and Wes are dependent on the electric potentialand ionic flux, therefore the mesoscopic model must account for the electrodiffusion dynamicsin the CED.In the CED, the asymmetric electrolyte solution is composed of multiple ionic species, thereexist Stern and diffuse electrical double layers at the surface of the electrodes and membrane,and the diffusion tensor is position dependent. These electrodiffusion dynamics can be modeledusing the GPNP continuum model given by:∂ci∂t = ∇ ·[Di · ∇ci − ziqciβDi · ∇φ+∇ ·Di · (ci∇ ln(1−N∑j=1NAa3jcj)], (4.3a)∇ · (ε∇φ) = −N∑i=1Fzici. (4.3b)In the generalized Nernst-Planck equation (4.3a), ci is the concentration, Di the diffusivitytensor, zi the charge valency, q the elementary charge, NA Avagadro’s number, and ai theeffective ion size of the chemical species i. Note that the expression in the [·] in (4.3a) is theconcentration flux J i. In the Poisson equation (4.3b), φ denotes the potential field, and FFaraday’s constant. The electrodiffusive model equation (4.3) is able to account for anisotropicdiffusion, asymmetric electrolytes, multiple ionic species, and the Stern and diffuse electricaldouble layers present at the surface of the electrodes and membrane. Note that if aj = 0 in(4.3), then (4.3) is equivalent to the standard Poisson-Nernst-Planck (PNP) model presented664.3. Dynamic Modelin Sec. 2.3.2.We must define the geometry and boundary conditions for equation (4.3). From MD simu-lations aqueous pores in the membrane are toroidal [167–169]. Therefore we consider the poregeometry given in Fig. 4.3 to compute the conductance. The material and boundary conditions׎௖௘׎௘ߝ௪ߝ௠Ω߲௠Ω௠Ω௥Ω௪݂ ݊ݎΩ߲௛௠Ω߲௪Ω߲௥Ω߲௖௘Ω߲௘௥݈௘݄௠݄௥݄௠݄/2Figure 4.3: Pore geometry used to compute the pore conductance Gp.of equation (4.3) for estimating J i and φ are given by:n · J i = 0 in ∂Ωm ∪ ∂Ωe ∪ ∂Ωec, (4.4)φm − φw = 0 in ∂Ωm,εm∇φm · n− εw∇φw · n = 0 in ∂Ωm, (4.5)Cs(φe − φ) + εwn · ∇φ = 0 in ∂Ωe,Cs(φec − φ) + εwn · ∇φ = 0 in ∂Ωec, (4.6)ci = cio in ∂Ωw, n · ∇φ = 0 in .∂Ωhm (4.7)Equation (4.4) states that the membrane surface is assumed to be perfectly polarizable (i.e.blocking). The spacers and tethers prevent any surface reactions at the bioelectronic interface;therefore, a no-flux boundary condition is present at the gold surface. To ensure the well-posedness of the Poisson equation in (4.3), the internal boundary conditions on the membraneto electrolyte interface are satisfied by (4.5) [80]. At the electrode surface a compact Sternlayer exists with a capacitance per unit area given by Cs. The Stern layer adjacent to theelectrodes is modeled using (4.6) with φe and φec the prescribed potentials at the respectiveelectrodes. (4.7) provides the ambient boundary conditions away from the pore with cio theinitial concentration, refer to Fig. 4.1.The membrane conductance Gp is computed using:Gp =IpVmIp = F∑ir∫0J i2pirdr, (4.8)674.3. Dynamic Modelwhere J i and Vm are computed using equation (4.3) with boundary conditions (4.4) to (4.7)in the geometry given in Fig. 4.1. To evaluate Wes, assuming negligible osmotic pressure, localmechanical equilibrium, and that pore expansion is isolated to the radial direction, the followingis used [51, 170, 171]:Wes(r) = −r∫0(∫Sn · (Tw − Tm)ndS)dr,Tw = εw(12 |∇φw|2I −∇φw ⊗∇φw),Tm = εm(12 |∇φm|2I −∇φm ⊗∇φm). (4.9)In (4.9), Tw and Tm are the Maxwell stress tensors [51, 171], I denotes the identity matrix, ndenotes the normal vector (Fig. 4.1), S the surface of the pore, and ⊗ is the dyadic product(i.e. ∇φ⊗∇φ = ∇φ∇φT ). The expression f = (Tw − Tm)n denotes the electrical force densityon the surface of the pore.To compare the numerical results of the GPNP model, the computation of Gp and Wes isalso performed using the EM and LM models presented below.Electroneutral Model of Pore Conductance Assuming electroneutrality (i.e.∑i zici = 0)and no steric effects ai = 0 [21], the governing equations of φw, the electrical potential in theelectrolyte solution Ωw defined in Fig. 4.3, can be derived by substituting the time derivativeof (4.3b) into (4.3a) for charge neutrality. The resulting elliptic equation is given by:∇ · (ς∇φ+∇κ) = 0ς =N∑i=1(qzi)2DikBTci, κ =N∑i=1qziDici, (4.10)with the parameters defined below (4.3). The boundary conditions of (4.10 at the electrodesurfaces ∂Ωe and ∂Ωce, and at the ambient boundary ∂Ωw, ∂Ωhm, and ∂Ωr are given by (4.6)and (4.7) respectively, refer to Fig. 4.3. In the membrane domain Ωm the electrostatic potentialφm is governed by Laplace’s equation for electrostatics ∇·(εm∇φ) = 0. The interface conditionsbetween the domains Ωw and Ωm are given by:n · ∇φw = 0 in ∂Ωm,φm = φw in ∂Ωm. (4.11)From the continuity of potential on ∂Ωm (4.11), there exists a surface charge on the membranegiven by ρs = εmn · ∇φm − εwn · ∇φw; therefore, the system of equations (4.10) with boundaryconditions (4.11) implicitly includes the membrane surface charge ρs [21, 51, 172].We denote the governing equations (3.4) with ai = 0 coupled with (4.10), the material684.3. Dynamic Modelparameters defined by (4.18) and (4.19), and the boundary conditions (4.4), (4.6), (4.7), and(4.11) as the Electroneutral Model (EM). Given the solution of the EM system of equations, theconductance Gp, (4.8), can be estimated.Electroneutral and Laplace Model of Electrical Energy Required to Form a PoreAssuming electroneutrality (i.e.∑i zici = 0) and no steric effects ai = 0, the electric potentialφ can be computed using the EM (4.10). Given φ from the EM, Wes can be computed using(4.9). If steady-state current (i.e. ∇ci = 0) is also assumed [51], the electrical potential φ isgoverned by Laplace’s equation ∇ · (ε∇φ) = 0 with ε defined by (4.19, and the interface andboundary conditions defined by (4.6), (4.7), and (4.11). We denote this as the Laplace Model(LM). Given φ from the LM, an estimate of Wes can be computed using the same method asfor the EM above.4.3.3 Macroscopic Fractional Order ModelThe fractional order macroscopic model (Fig. 4.1) links the experimentally measured currentto the mesoscopic model via the pore conductance Gp and electrical energy required to form apore Wes, and the CGMD model via the surface tension σ and line tension γ. The macroscopicmodel of the CED is constructed by making asymptotic approximations to the Smoluchowski-Einstein equation [41, 45, 46] and coupling the result with a system of nonlinear fractional orderdifferential equations that model the surface and electrolyte dynamics.At the electrode to electrolyte interface there exists electrical double layers (i.e. Stern layerand diffuse charge layer). For a planar electrode with no defects or tethers, this interface canbe modeled using an ideal capacitor. However, as a result of the electrode surface containingdefects from the manufacturing process, and the spacers and tethers bound to the electrodesurface, there may exist diffusion-limited processes including: charge transfer, reaction limitedcharge transfer, and adsorption [173]. In the macroscopic model, these processes are modeledusing a fractional capacitor (i.e. the capacitor voltage is related, by a fractional order differentialoperator, to the current traveling through the capacitor). Ctdl, and Cbdl denote the fractionalorder capacitances for the counter electrode and electrode respectively. The bulk electrolytesolution is modeled as completely ohmic with resistance Re. The charging dynamics of themembrane is modeled by the capacitance Cm. The tethered membrane conductance Gm(t, Vm)is both time and membrane voltage dependent with Vm denoting the transmembrane poten-tial. The dependency of Gm is a result of the process of electroporation that takes place togenerate/destroy aqueous pores in the membrane. The excitation potential Vs(t) applied acrossthe two electrodes closes the circuit. The equivalent circuit model of the tethered membranedevice is given in Fig. 4.1. The governing equations for the lumped circuit model of the tetheredmembrane (Fig. 4.1) are given by (4.14).The membrane conductance Gm is modeled using asymptotic approximations to the694.3. Dynamic ModelSmoluchowski-Einstein (ASE) equation for electroporation. The ASE links the pore conduc-tance Gp to the conductance of the membrane and is given by:Gm =bN(t)c∑i=1Gp(ri),dridt = −DkBT∂W (ri, Vm)∂rifor ri ∈ {1, 2, . . . , bN(t)c},W = 2piγri − piσr2i + (Cri)4 +Wes + 0.5Ktr2i ,dNdt = αe( VmVep )2(1− NNoe−q(VmVep )2). (4.12)In (4.12), α is the pore creation rate coefficient, Vep is the characteristic voltage of electropo-ration, No is the equilibrium pore density at Vm = 0, and q = (rm/r∗)2 is the squared ratio ofthe minimum energy radius rm at Vm = 0 with r∗ the minimum energy radius of hydrophilicpores [96, 174–176]. W is the free energy of a hydrophilic pore which consists of the pore edgeenergy γ, the membrane surface tension σ, the electrostatic interaction between lipid heads,and Wes (4.9), and the energy contribution from the tethers with Kt denoting the the associ-ated spring constant [48, 51–54]. Note, the energy model for tethers is identical to that of thecytoskeletal network presented in [177–180]. Intuitively, W is the free energy change of the lipidmembrane necessary for the formation of an aqeous pore. For example, W can be estimatedfrom molecular dynamics simulations by pulling lipids in the membrane to form a pore, referto [169] for details.The derivation of (4.12) is based on making physiologically relevant approximations tothe Smoluchowski-Einstein equation for electroporation. The Smoluchowski-Einstein equationgoverns the distribution of pores as a function of their radius r and time t [41, 45, 46]. Ifwe denote n(r, t) as the pore density distribution function, then the Smoluchowski-Einsteinequation is given by:∂n∂t = D∂∂r[ nkBT∂W∂r +∂n∂r]+ S(r), (4.13)where D is the diffusion coefficient of pores, kB is the Boltzmann constant, T is the temperature,W is the pore energy, and S(r) models the creation and destruction rate of pores. S(r) accountsfor the hydrophilic to hydrophobic transition energy involved in pore formation with detaileddiscussion provided in [41, 53, 54, 181]. Making the physiologically relevant assumption thatdiffusion term (i.e. ∂n/∂r) in (4.13) is negligible, and the characteristic time scale of W is longerthen 0.1 µs, then the process of electroporation can be modelled by (4.12). Note that (4.12) hasbeen used by several authors for modelling DNA translocation into cells [43, 44, 96, 174–176].To link the CGMD and GPNP results with experimental measurements, the fractional order704.4. Numerical and Experimental Resultsmacroscopic model is used. The macroscopic model is given by (Fig. 4.1):dVmdt = −(1CmRe+ GmCm)Vm −1CmReVdl +1CmReVs,dpVdldtp = −1CdlReVm −1CdlReVdl +1CdlReVs,I(t) = 1Re(Vs − Vm − Vdl), (4.14)where Cdl is the total capacitance of Ctdl and Cbdl in series with p denoting the order of thefractional order operator. Given the drive potential Vs(t), and the static circuit parametersCtdl, Cbdl, Cm, and electrolyte resistance Re, the membrane conductance Gm(Gp,Wes, σ, γ,N)can be estimated from the measured current I(t). The dynamics of Gm is modeled usingasymptotic approximations to the Smoluchowski-Einstein given by (4.12). The parametersfrom the CGMD and continuum model are linked to the membrane conductance Gm which canbe measured experimentally from the current I(t) using equation (4.14).4.4 Numerical and Experimental Results4.4.1 Structure and Biomechanics of CEDImportant parameters related to the structure and biomechanics of the controlled electropora-tion measurement device can be computed using the CGMD model (Fig. 4.1). Here the diffusionD, membrane thickness hm, surface tension σ, and line tension γ are computed.The diffusion coefficient of the DphPC and GDPE lipids, and water is provided in Table 4.1.The computed diffusion of the DphPC and GDPE lipids in the proximal (i.e. adjacent to thetethering reservoir) layer and distal (i.e. adjacent to the bulk water) are nearly identical. Thediffusion of DphPC is related to that of GDPE by a multiplicative factor of 1.13. The effects ofthe tethers cause the diffusion coefficient of DphPC to decrease by a factor of approximately 3.3in the proximal layer, and 2.3 in the distal layer. Similarly for GDPE the decrease is 2.6 in theproximal layer, and 2.1 in the distal layer. This result is in agreement with the experimentalresults [182, 183] for different lipids and tethering densities. The diffusion of water in the bulkregion is related to the tethering reservoir diffusion by a factor of approximately 1.5 for the25% tethered membrane. Interestingly the diffusion of water in the tethering reservoir is 2.7nm2/ns and in the bulk is 1.9 nm2/ns for the 0% tethered membrane. The interplay betweenhydrogen bond breaking and cooperative rearrangement of regions of approximately 1 nm in sizecause the diffusion to significantly increase in nanoconfined water regions [184]. Since explicithydrogen bonds are not included in the CGMD water [74], we attribute the increased diffusioncoefficient to the cooperative rearrangement of water molecules. A unique result of this study isthat no anomalous diffusion was detected for water and lipid headgroups near the outer surface714.4. Numerical and Experimental Resultsof bilayer lipid membrane. This is in contrast to other dynamical studies [173, 185–187] ondifferent membranes which report an anomalous water and lipid diffusivity in the vicinity ofthe membrane surface.Tethering density 0% 25%Proximal Layer DphPC 0.288±0.002 0.086±0.001GDPE 0.253±0.001 0.109±0.001Distal Layer DphPC 0.285±0.002 0.111±0.001GDPE 0.252±0.001 0.120±0.001Bulk Water 1.921±0.041 1.746±0.019Tethering Reservoir Water 2.710±0.321 1.126±0.038Table 4.1: Diffusion Coefficient D (nm2/ns)To compute hm the particle density of the lipid headgroups is used (refer to Sec. 4.4.9for details). The thickness of the 25% and 0% tethered membrane is hm = 3.53 nm andhm = 3.48 respectively. The thickness of the phytanyl tails (i.e. hydrocarbon tails) was alsocomputed for the 25% and 0% tethered membranes and is 2.15 nm and 2.11 nm respectively.This is in agreement with the experimentally measured thickness for DphPC based tetheredmembranes [8].The surface tension and line tension are computed using the method presented in the Sup-plementary Information. The resulting values are: σ = 15 mN/m, and γ = 12 pN. These valuesare in agreement with the experimental results [188] and simulation results [169, 189, 190]reported in the literature for similar DphPC based membranes.4.4.2 Lipid Energetics and Pore DensityThe potential of mean force (PMF) of lipids in a lipid bilayer is a key thermodynamic propertythat can be used to estimate the pore density and the free energy of lipid flip-flop. Lipid flip-flop occurs when a lipid molecule flips from one side of the lipid bilayer to the other. Here theumbrella sampling method is used to compute the PMF for moving a single DphPC lipid alongthe normal direction to the membrane surface for both the 0% and 25% tethered membranes.The snapshots for selected umbrella simulation windows for the 25% tethered membrane areshown in Fig. 4.4.The computed PMF for both tethering densities is provided in Fig. 4.5(a). As expected, theminimum of the PMFs at -1.82 nm and 1.82 nm corresponds to the equilibrium position of theDphPC lipid. There is an increase in the PMF from these equilibrium positions as a result of thehydrophobic tail coming into contact with the water, and the hydrophilic headgroups cominginto contact with the hydrophobic phytanyl tails of the adjacent lipids. The difference in energybetween the equilibrium position and when fully solvated in the water solution correspondsto the free energy of lipids desorption. The free energy of desorption for the 0% and 25%724.4. Numerical and Experimental Resultsa b cFigure 4.4: Snapshots of restrained DphPC lipid in 25% DphPC membrane for umbrella sam-pling. (a) is the lipid in the equilibrium position, (b) the lipid at the center of the bilayer, and(c) the lipid in bulk water. Water is represented by light blue beads, pulled lipid as magentaspheres, lipid tails as green lines, DphPC and GDPE headgroups NC3,PO4,OH as blue,orangeand red balls, tethers as violet sticks, and spacers as tan sticks. The gold is represented by theyellow beads.membrane is 85.93±0.5 kJ/mol and 91±0.5 kJ/mol respectively. As expected, the tethers causethe associated energy of lipid desorption to increase as compared with the untethered membrane.As the lipid is moved to the center of the membrane, the steep slope in the PMF is causedby water and the lipid headgroups interacting with the bilayer interior. This characteristicof the PMF has been observed when other charged/polar molecules are transferred into thehydrophobic interior of the membrane [191–194]. This effect can be viewed in Fig. 4.5(b)-Fig. 4.5(c) and is associated with the formation of an aqueous pore.Remark: The computed PMF for the 0% and 25% energy for lipid flip-flop provided inFig. 4.5(a) is consistent with pore formation when other charged/polar molecules are are trans-ferred in to the hydrophobic interior of a membrane using different force field and methods [191–194]. However, only the formation of transient pores is observed from the CGMD simulations.This can be attributed to lack of electrostatic interaction in CGMD water model. CGMDwater has a zero dipole moment so simulations are run with a dielectric constant of 15 for im-plicit screening of electrostatic interactions. This means that the interaction of polar/chargedmolecules is under-estimated in hydrophobic environments with MARTINI water model (e.g.when a lipid (PO4 group) headgroup is placed in the interior of a lipid bilayer).Using the PMF (Fig. 4.5(a)), the free energy required for complete flip-flop of a single lipidis equal to the energy required to move a lipid from it’s equilibrium position to the center ofthe bilayer, then to the other leaflet’s equilibrium position. Therefore, from the maxima inthe PMF’s between equilibrium and the bilayer center, the free energy barrier for flip-flop ofthe DphPC lipid increases from 89 kJ/mol for the free bilayer to 103.17 kJ/mol for the 25%734.4. Numerical and Experimental Resultstethered membrane. These results show that inclusion of tethers in the DphPC membraneprevents the formation of defects and increases the free energy for the translocation of chargedheadgroups across the membrane. Assuming the energy required to form a pore, denoted by∆Gp, is equal to the energy require for lipid flip-flop, the equilibrium pore density can becomputed from ρ0 = e−β∆Gp/AL, where AL is the area per lipid [191]. For DphPC AL = 0.69nm2, therefore the associated equilibrium pore density for the 0% and 25% tethered membranesis 4375 pores/m2 and 21 pores/m2 respectively. This is approximately six orders of magnitudeless then the experimentally measured pore density which is in the range of 1.5× 108 to 3× 108pores/m2. A possible cause for this discrepancy is that the formed membrane contains defectsresulting from the manufacturing process, there are not included in the CGMD model.Figure 4.5: (a) is the associated PMFs for a 0% and 25% DphPC lipid membrane. (b) and (c)provide the CGMD simulation snapshots of the restrained DphPC lipid at the center of themembrane used to construct the PMF for the 0% and 25% tethered DphPC respectively. Wateris represented by light blue beads, pulled lipid as magenta spheres, lipid tails as green lines,DphPC and GDPE headgroups NC3,PO4,OH as blue,orange and red balls, tethers as violetsticks, and spacers as tan sticks.4.4.3 Pore Conductance and Electrical Energy Required to form a PoreTo interpret the experiment measurements with the CGMD results requires an estimate of thepore conductance Gp and electrical energy required to form a pore Wes. To gain insight into theeffects the diffusion parameter, and assumptions of negligible steric effects, electroneutrality, andnegligible ion gradients have on the computed Gp and Wes, numerical estimates are computedusing the GPNP, PNP, EM, and LM models presented in Sec. 4.3.2.744.4. Numerical and Experimental ResultsIn Fig. 4.6 the estimated pore conductance computed using the GPNP, PNP, and EMmodels is presented. As seen the pore conductance predicted using the GPNP follows a Gp ∝r relationship. For membranes with sufficiently large electrolyte baths and pore radii (i.e.electrolyte bath is hundreds of nm thick and r > tm), the pore conductance follows Gp ∝r [21, 42, 45, 66], in agreement with the spreading conductance derived from Laplace’s equationin [65, 195]. Note that the effect Gp ∝ r for r < tm and hr = 4 nm is only predicted whenthe effects caused by asymmetric electrolytes, finite ion size, and Stern and diffuse layers areaccounted for. In Fig. 4.6(a), the GPNP and PNP models produce differing conductanceestimates as a result of the steric effects present. Recall that for ∑Ni=1NAa3i ci  1 the stericeffects are negligible and the estimated conductance using the GPNP and PNP models would beidentical. As seen by comparing the estimated conductance Gp in Fig. 4.6(a), the assumption ofelectroneutrality causes a noticeable decrease in the computed conductance Gp. As mentionedin the introduction, the pore conductance Gp may be dominated by the spreading conductance,which follows a Gp ∝ r proportionality, when the electrolyte solution is sufficiently geometricallyconstrained. From Fig. 4.3, the tethering reservoir is hr = 4 nm, and from Fig. 4.6(a) we seethat Gp ∝ r; therefore, we conclude that the conductance of an aqueous pore in the engineeredtethered membrane is dominated by the spreading conductance. As the diffusivity in thetethering reservoir, Dr, decreases the pore conductance decreases, as seen in Fig. 4.6(b). Thisis expected as less ions can flow through the pore as a result of reduced ion mobility.BAa bFigure 4.6: Numerically predicted pore conductance Gp, defined in (4.8). (a) provides thepredicted pore conductance Gp computed using the GPNP, PNP, and EM models presented inSec. 4.3.2. (b) is the predicted Gp for different tethering reservoir diffusivities. The geometryof the pore is given in Fig. 4.3 with the parameters of the governing equations and boundaryconditions provided in Table 4.2.Fig. 4.7 compares the computed electrical energy required to form a pore using the GPNP,PNP, EM, and LM defined in Sec. 4.3.2. For small pores below 1 nm all the models providesimilar estimated forWes, as seen in Fig. 4.7(a). The PNP and EM models provide a significantlylower estimate of Wes compared to the GPNP and LM models for large pore radii above 4 nm.The discrepancy between the estimated Wes is a result of the assumption of negligible steric754.4. Numerical and Experimental Resultseffects in the PNP model, and the assumption of negligible steric effects and electroneutralityin the EM model. Note that although the GPNP and LM models provide similar predictionsof Wes, the LM assumes negligible steric effects, electroneutrality, and steady-state current (i.e.∇ci = 0) which results in the estimated voltage distribution on the surface of the membrane todiffer with the voltage distribution predicted from the GPNP. Qualitatively at the surface of themembrane the GPNP model has the interface condition (4.5) such that εm∇φm ·n = εw∇φw ·n;however, the interface condition for the LM model (4.11) causes εm∇φm · n 6= εw∇φw · n onthe surface. This results in the LM overestimating the voltage potential when compared withthe GPNP. From (4.9), the overestimated potential causes the computed Wes to be larger whenusing the LM model as compared with the GPNP model. As discussed, the assumption ofWes(r, Vm) ∝ V 2m is typically invoked to simplify the computation of Wes(r, Vm) [42, 51, 196].From Fig. 4.7(b), we compute Wes(r, Vm) explicitly for several transmembrane potentials andfind that the proportionality follows a fractional power law. This illustrates the importance ofincluding effects caused be electrodiffusion. As illustrated in Fig. 4.7(c), reducing the diffusioncoefficient in the tethering reservoir Dr causes a slight reduction in the estimated Wes. Incomparing Fig. 4.7(b) with Fig. 4.7(c), the main contribution to the change in Wes results froma change in transmembrane potential. BACa bcFigure 4.7: Numerically predicted electrical energy Wes, (4.9), required to form an aqueouspore. (a) compares the predicted Wes computed using the GPNP, PNP, EM, and LM modelsdefined in Sec. 4.3.2 for the transmembrane potential of Vm = 500 mV. (b) presents estimates ofWes computed using the GPNP for the transmembrane potentials listed. (c) provides estimatesof Wes computed using the GPNP for Vm = 500 mV for different tether reservoir diffusivities.The parameters of the governing equations and boundary conditions can be found in Table 4.2.764.4. Numerical and Experimental ResultsFinally the CGMD diffusion coefficients in Table 4.1 are used to compute Gp and Wes withthe results presented in Fig. 4.8. As seen, the computed conductance is in agreement withthe experimentally measured conductance of a single pore obtained from planar bilayer mem-branes (BLMs) using patch-clamp and linearly rising current protocols [5–7]. As expected,for pore radii r > 3 nm the experimentally measured conductance from the BLMs is largerthen for the CED as the electrode surface in the BLMs is not in close proximity (i.e. 4 nm)to the membrane surface in the BLM experimental setup [5–7]. From Fig. 4.8(a), the com-puted conductance approximately follows a Gp ∝ r relationship as a result of the “spreadingconductance” [21, 45, 65, 197]. Fig. 4.8(b) presents the numerically computed Wes using theresults from the CGMD diffusion coefficients. The results compare favorably with previouslycomputed Wes(r, Vm) using simplified governing equations that do not include electrodiffusiveeffects [51, 196]. From Fig. 4.8(b), the proportionality between Wes and Vm follows a fractionalpower law.Figure 4.8: Numerically predicted Gp and Wes computed using the GPNP model in Sec. 4.3.2with parameters defined in Table 4.1 and Table 4.2. The experimentally measured conductance(BLMs) is obtained from [5–7].4.4.4 Electrode Surface Impedance and DynamicsIn this section experimental measurements are used to establish that a diffusion-limited processis present at the surface of the gold electrodes of the CED, and that the process can be accountedfor using a fractional order model. At the surface of the gold electrode there is a self-assembledmonolayer of spacer and tether molecules. To investigate if a diffusion-limited process is presentat the electrode interface, a gold surface was constructed with only spacer molecules present,denoted as the spacer surface. The fractional order macroscopic model of the spacer surface isgiven by:dpVdldtp = −1CdlReVdl +1CdlReVs (4.15)with the bioelectronic interface modeled using a constant-phase-element composed of a capac-itance Cdl and the fractional order operator p, Vdl is the voltage drop across the bioelectronicinterface, Re is the electrolyte resistance, and Vs is the drive potential. Note that (4.15) is774.4. Numerical and Experimental Resultsequivalent to the fractional order model of the CED (4.14) with no tethered membrane present.To detect if a diffusion-limited process is present impedance measurements of the spacer sur-face were performed. For a sinusoidal excitation Vs(t) = Vo sin(2pift) with frequency f andmagnitude Vo below 50 mV, the impedance of the spacer surface is given by:Z(f) = Re +1(j2pif)pCdl, (4.16)where f is the frequency of excitation with the parameters Re, p, and Cdl defined in (4.15). Asdiscussion in Sec. 2.3.1, if p < 1 then a diffusion-limited process is present, and if p = 1 then adiffusion-limited process is not present.In Fig. 4.9, the experimentally measured and numerically predicted impedance for the spaceronly electrode is presented for electrolyte concentrations of 2 M, 1 M, 500 mM, 200 mM NaCl.As expected, as the saline concentration increases, the electrolyte resistance Re decreases. Ifa diffusion-limited process is not present then ∠Z(f) ≈ 90◦ for f < 1 Hz. As seen fromFig. 4.18(a) ∠Z(f) ≈ 76◦ for f < 1 Hz, therefore for all concentrations measured there is adiffusion-limited process present. The estimated fractional order parameter is p = 0.86.10−1 100 101 102 1036065707580∠Z(f) [o ]f [Hz] 10−1 100 101 102 103103104105106Z(f) [Ω]f [Hz]2 M200 mMFigure 4.9: The measured and predicted impedance of the spacer surface. All predictions arecomputed using (4.16) with the parameters defined in Table 4.3.To validate the macroscopic model (4.15), Fig. 4.10 provides the experimentally measuredand numerically predicted current response of the spacer only electrode for a drive potentialVs defined by a 100 V/s rise for 5 ms, and a -100 V/s fall for 5 ms. As seen, the fractional-order differential equation (4.15) with p = 0.83 provides an accurate prediction of the currentresponse. The fractional-order behavior at the electrode surface may be caused by: diffusion-limited charge transfer, reaction limited charge transfer, diffusion-limited adsorption on theelectrode [173]. Though the source of the behavior is unknown, the dynamics of the interfacecan be modeled using the fractional order differential equation (4.14) as illustrated in Fig. 4.9and 4.10.784.4. Numerical and Experimental Results0 1 2 3 4 5 6 7 8 9 10−10−50510Time [ms]Current[µA]  ExperimentalPredictedFigure 4.10: The measured and predicted current response of the spacer surface. All predictionsare computed using (4.15) with the parameters defined in Table 4.3.4.4.5 Quality of Formed Membrane via Impedance MeasurementsTo detect if the membrane contains significant defects, impedance measurements can be used.Possible membrane defects include patches with the gold electrode directly exposed to the bulkelectrolyte, or with portions of bilayer sandwiched together. The defect density in the mem-brane can be estimated from the impedance measurements using the protocol presented in [102].If the estimated capacitance of the membrane significantly increases this indicates that eitherelectrodesorption of the tethers and spacers has occurred, or portions of the tethered membranehave been released into the electrolyte. If the estimated equilibrium pore conductance signif-icantly increases then catastrophic voltage breakdown of the membrane has occurred causingseparated areas of the membrane to degrade. Typical values for membrane capacitance andconductance are 0.5-1.3 µF/cm2 and 0.5-2.0 µS for an intact 1%-100% tethered membrane withsurface area 2.1 mm2. For all experimental measurements, the membrane contained negligibledefects at the start and end of the trials.As an example, the quality of the 1% and 10% tethered membrane is presented. Using thefractional order model of the CED (4.14), for a sinusoidal excitation Vs(t) = Vo sin(2pift) withmagnitude Vo = 20 mV and frequency f , the impedance of the CED is given by:Z(f) = Re +1Go + j2pifCm+ 1(j2pif)pCdl. (4.17)The circuit parameters in (4.17) are defined above equation (4.14), and j denotes the complexnumber√−1. Fig. 4.16 and Fig. 4.12 present the experimentally measured and numericallypredicted impedance for the 1% and 10% tethered membranes respectively. As seen, the pre-dicted impedance is in excellent agreement with the experimental measured impedance and isconsistent with a membrane containing negligible defects.794.4. Numerical and Experimental Resultsf [Hz]10-11001011021036Z(f)[o]30405060708090f [Hz]10-1100101102103Z(f)[+]104105106107ExperimentalPredictedFigure 4.11: The measured and predicted impedance of the 10% tether density DphPC bilayermembrane. All predictions are computed using (4.17) with the parameters defined in Table 4.3.f [Hz]10-11001011021036Z(f)[o]30405060708090f [Hz]10-1100101102103Z(f)[+]104105106107ExperimentalPredictedFigure 4.12: The measured and predicted impedance of the 1% tether density DphPC bilayermembrane. All predictions are computed using (4.17) with the parameters defined in Table 4.3.To ensure the formed S. cerevisiae and E. coli tethered membranes contain negligible defectsimpedance measurements are made. The results are provided in Fig. 4.13 and Fig. 4.14. As seen,excellent agreement between the experimentally measured and numerically predicted impedanceis obtained indicating that the membrane contains negligible defects.10−1 100 101 102 10330405060708090∠Z(f) [o ]f [Hz] 10−1 100 101 102 103103104105106107f [Hz]Z(f) [Ω]  S. cerevisiaeE. coliFigure 4.13: The measured and predicted impedance of the 1% tether density S. cerevisiae andE. coli membranes. All predictions are computed using (4.17) with the parameters defined inTable 4.3.804.4. Numerical and Experimental Results10−1 100 101 102 10330405060708090∠Z(f) [o ]f [Hz] 10−1 100 101 102 103104105106107f [Hz]Z(f) [Ω]  S. cerevisiaeE. coliFigure 4.14: The measured and predicted impedance of the 10% tether density S. cerevisiaeand E. coli membranes. All predictions are computed using (4.17) with the parameters definedin Table 4.3.4.4.6 Measured Dynamics of Controlled Electroporation MeasurementDeviceTo characterize the predictive accuracy of the atomistic-to-observable model (Fig. 4.1), exper-imental measurements are used. Fig. 4.15(a) to 4.15(d) presents the results for 1%, 10%, and100% tethering densities and various drive potentials. As expected, the effects of electropora-tion are negligible for drive potentials below 250 mV as seen in Fig. 4.15(a) and Fig. 4.15(b).As suggested from the CGMD results, the higher the tethering density the more resistant themembrane is to the effects of electroporation. As seen in Fig. 4.15(a) and Fig. 4.15(b), thiseffect is pronounced for the larger drive potentials. Fig. 4.15(c) shows that the 100% teth-ered membrane can withstand drive potentials of up to 800 mV. To ensure the membrane wasnot irreversibly damaged, the current response of the membrane was compared to that of thepredicted model for potential steps of 50 mV, 100 mV, 150 mV, and 200 mV. Fig. 4.15(d)shows excellent agreement between the experimentally measured current and predicted currentsuggesting negligible defects are present.814.4. Numerical and Experimental Resultsa bc dDphPC DphPC100% DphPC100% DphPCFigure 4.15: Panels (a) to (d) provide the experimentally measured and numerically predictedcurrent response I(t) for 1%, 10%, and 100% tethered DphPC membranes. In (a) the drivepotential Vs(t) is defined by a 1 ms linearly increasing potential with a slope of 10 V/s, 40 V/s,70 V/s, and 90 V/s followed by a 1 ms linearly decreasing potential with identical slope. In (b)Vs(t) is a 5 ms linearly increasing potential with slope of 50 V/s, 200 V/s, 300 V/s, and 450V/s followed by a 5 ms linearly decreasing potential with identical slope. In (c) Vs(t) is definedby a 8 ms linearly increasing potential with a slope of 100 V/s followed by a 8 ms decreasingpotential with identical slope. In (d) Vs(t) is a step of 50 mV, 100 mV, 150 mV, and 200 mV.All predictions are computed using equation (4.14) with the parameters defined in Table 4.3.4.4.7 Pore Population and DynamicsBelow the predicted pore conductance Gp, (4.8), electrical energy Wes, (4.9), and the elec-troporation model given by (4.14) and (4.12) are used to predict the current response of theengineered tethered membrane. The predicted current response is compared to experimentallymeasured data to validate the accuracy of the model. Note that all electroporation processeswere reversible and did not cause permanent damage to the membrane.From (4.14) and (4.12), if the drive potential Vs is applied and the resulting current is Is;then if the drive potential −Vs is applied the resulting current must be −Is if only the process ofelectroporation is present. For all the tethering densities and membrane compositions tested thisrelation was observed in all experimental current measurements and therefore we concluded thatthe only process present is that of electroporation. The drive potential Vs(t) used to producethe results in Fig. 4.16 is defined by a linearly increasing potential of 100 V/s for 5 ms proceededby a linearly decreasing potential of -100 V/s for 5ms.The experimental measurement and predicted voltages, pore radii, membrane resistance,824.4. Numerical and Experimental Resultsand current are presented in Fig. 4.16 for the 10% tethered DphPC bilayer membrane. FromFig. 4.16(a), the experimentally measured and numerically predicted current are in excellentagreement. As seen in Fig. 4.16(b), the application of the voltage excitation immediatelycauses an increase in the double-layer voltage Vdl as a result of the charge increase in the chargedistribution at the electrode surface. The transmembrane potential Vm simultaneously increasesas a result of the excitation potential. The increase in Vm results in the formation of pores. Asseen in Fig. 4.16(c), a dramatic change in the resistance results after the application of the drivepotential. In Fig. 4.16(d) the maximum radius rmax and mean radius r are provided to illustratethe spread in pore radii. As Vm increases, pores are generated and expand according to (4.12).From (4.12), all pores diffuse to the minimum-energy pore radius given by ∂W/∂ri = 0 withan advection velocity proportional to D/kBT . As seen in Fig. 4.16(d), generated pores rapidlyexpand to the minimum-energy pore radius as the spread between rmax and r is negligible.This allows the number of pores N , from (4.12), to be computed using the relation N =1/(RmGp(rmax)) with Gp given in Fig. 4.6(a).DBACa bc dFigure 4.16: The measured and predicted current, voltage potentials, membrane resistance,and pore radii for the drive potential Vs(t), defined at the beginning of this section, for the10% tether density DphPC bilayer membrane. (a) is the measured and predicted current, (b)the predicted transmembrane Vm and double-layer potential Vdl defined in (4.14), (c) is theestimated membrane resistance, and (d) the estimated maximum rmax radius, and mean poreradius r. All predictions are computed using (4.14) and (4.12) with the parameters defined inTable 4.3.834.4. Numerical and Experimental Results4.4.8 Effect of Variations in Membrane Composition and Tether DensityTo gain insight into the process of electroporation and validate the molecular dynamics resultsin [160, 161], experimental measurements and numerical results for CEDs composed of DphPC,S. cerevisae, and E. coli lipids are presented in this section.Fig. 4.17 provides the experimental current measurement and the predicted current andmembrane resistance Rm = 1/Gm for the 1% and 10% DphPC bilayer, and the 100% DphPCmonolayer membrane. In comparing the resulting current between the 1%, 10%, and 100%tethered case, Fig. 4.17(a), we see that as the tethering density increases the effects of elec-troporation decreases. This is an expected result as the tethers provide structural supporthindering the nucleation of pores reducing the equilibrium pore density No and increasing thecharacteristic voltage of electroporation Vep. As seen in Fig. 4.17(b), the resistance begins tochange at approximately 1 ms when the transmembrane potential reaches a sufficiently high tocause the nucleation of pores. The estimated spring constant Kt for the 1%, 10%, and 100%tethering density are: 0 mN/m, 2±0.5 mN/m, and 20±4 mN/m. For the 1% tether density thespring constant is negligible as expected. For the 100% tethering case pores cannot expand asa result of the spring constant Kt, therefore the decrease in resistance is primarily a result ofpore nucleation and destruction governed by (4.12). For the 100% tether density membrane, itmay be the case that all pores in the membrane are hydrophilic as the tethers may prevent thetransition from the hydrophilic to hydrophobic structure. If only hydrophilic pores are present,the membrane resistance would be dominated by the nucleation of pores and not the dynamicsof the pores. Note that the molecular structure of the aqueous pores cannot be reliably inferredusing continuum theory models and would require the use of molecular dynamics or similarnon-continuum models. Interestingly, for the 1% membrane structures the resistance begins todecrease at 9.2 ms, and for the 10% membrane at 9.4 ms after the initial application of thedrive potential Vs(t) defined at the beginning of this section. This is a result of the charge ac-cumulation in the electrical double layers at the gold electrode surface, Vdl, discharging causingan increase in the magnitude of the transmembrane potential Vm. This illustrates the impor-tance of including electrical double-layer effects when modeling gold electrodes. Note that whenusing (4.14) and (4.12) for estimating the effects of electroporation for rapidly changing drivepotentials, the double-layer capacitance in (4.14) can become time dependent [63]. In suchcases the dynamics of the time dependent capacitance can be estimated using the GPNP modeldefined in (4.3) using the method outlined in [63] with the electroporation model developed inthis chapter. The thickness of the membrane can be estimated using hm = εmAm/Cm withAm = 1.2 mm2, the area of the membrane surface, and εm and Cm given in Table 4.2 andTable 4.3. For the 1%, 10%, and 100% membranes we obtain a thickness of: 3.54 nm, 3.54 nm,3.40 nm. These values are in excellent agreement with the results of Neutron Reflectometrymeasurements of similar DphPC based tethered membranes [8]. As seen, the thickness of the844.4. Numerical and Experimental Resultstethered DphPC membrane is approximately constant between the 1% and 10% tether den-sities. The 100% DphPC monolayer is slightly thinner then the 1% and 10% DphPC bilayermembrane. The reduction in thickness between the 100%, and the 1% and 10% is a result of thecombined effect of an increased tether density and the dibenzyl group that binds the phytanyltails in the tethered DphPC monolayer. BAa bFigure 4.17: Experimentally measured and numerically predicted current I(t) (a), and mem-brane resistance Rm = 1/Gm (b) for the drive potential Vs(t) is defined by a linearly increasingpotential of 100 V/s for 5 ms proceeded by a linearly decreasing potential of -100 V/s for 5ms.The tethering densities 1% and 10% correspond to the DphPC bilayer and the 100% corre-sponds to the DphPC monolayer. All predictions are computed using (4.14) and (4.12) withthe parameters defined in Table 4.3.In Fig. 4.18, the experimentally measured and the current obtained from our two-levelpredictive model I(t) is displayed for several different linearly increasing and decreasing drivepotentials. As seen from Fig. 4.18(a)-(d), there is excellent agreement between the experimen-tally measured and numerically predicted current. For small magnitude drive potentials onewould expect the membrane resistance to remain constant as the effects of electroporation,governed by (4.14) and (4.12), are negligible. Indeed from Fig. 4.18(a) and Fig. 4.18(d), we seethat the electroporation effects are negligible for drive potentials from 50 to 80 V/s for the 1 msrise, and 10-40 V/s for the 5 ms rise. The reason the 5 ms rise, Fig. 4.18(a) and Fig. 4.18(c),has larger relative electroporation effects as compared with the 1 ms rise, Fig. 4.18(b) andFig. 4.18(d), is that the nucleation and dynamics of pore radii evolve for a longer period oftime at a sufficiently high transmembrane potential. As expected, the magnitude of the currentresponse for the 10% tethered membrane, Fig. 4.18(a) and Fig. 4.18(b), is less then the currentresponse for the 10% tethered membrane, Fig. 4.18(c) and Fig. 4.18(d), as a result of the tethershindering the nucleation and expansion of pores.854.4. Numerical and Experimental ResultsDBACa bc dFigure 4.18: Experimentally measured and numerically predicted current response I(t) for the10% tethering density DphPC membrane, panels (a) and (b), and the 1% tether density DphPCmembrane, panels (c) and (d). In panels (a) and (c), the drive potential Vs(t) is defined by a1 ms linearly increasing with a rise time of 50 to 500 V/s in steps of 50 V/s proceeded by alinearly decreasing potential of -50 to -500 V/s in steps of -50 V/s for 1 ms. In panels (b) and(d), the drive potential Vs(t) is defined by a 5 ms linearly increasing potential for 10 to 100V/s in steps of 10 V/s proceeded by a linearly decreasing potential of -10 to -100 V/s in stepsof -10 V/s for 5 ms. The numerical predictions are computed using (4.14) and (4.12) with theparameters defined in Table 4.3.Fig. 4.19 presents the experimentally measured and numerically predicted current responsefor 1% and 10% tetherd DphPC, E. coli, and S. cerevisae membranes. As seen, excellent agree-ment is obtained between the predicted and measured current response. This allows the model(Fig. 4.1) with experimental measurements from the CED to be used to estimate importantbiological parameters for the process of electroporation in unique membrane architectures. Con-sider the relative dielectric permittivity εm of the tethered DphPC, E. Coli, and S. cerevisiaemembranes. For homogeneous hydrocarbon tails the permittivity is approximately 2, howeverthe lipid headgroups cause the effective permittivity εm to be larger. For the 10% tetheredDphPC membrane hm = 3.5 nm, Am = 2.1 mm2, and Cm is the range of 12.5 nF to 15.5 nF.Therefore the relative permittivity is εm = 2.35 − 2.92. For the 1% tethered DphPC Cm is inthe range of 16 nF to 17.5 nF, and assuming the thickness of identical to that of the 0% tetheredDphPC hm = 3.4 nm, the relative permittivity is εm is in the range 2.92 to 3.20. Note that therelative permittivity of the membrane εm must be between the values of 2 for pure hydrocar-bon and 80 for an electrolyte at physiological concentrations. Therefore as the density of waterincreases between the distal and proximal layer, or the hydrocarbon thickness decreases it is864.4. Numerical and Experimental Resultsexpected that the relative permittivity will increase. Using the CGMD results Sec. 4.4.1 thedecrease in hm between the tethered and untethered DphPC membrane is a result of changesin the thickness of the hydrocarbon region. Therefore the decrease in εm between the 10% and1% tethered DphPC membranes results from changes in the hydrocarbon thickness. This showsthat the dielectric permittivity is dependent on the tether density. To estimate the permittivityof the tethered E. coli and S. cerevisae membranes we chose hm = 3.29 nm for the E. coli, andhm = 4.30 nm for the S. cerevisae as justified by the molecular dynamics results [161, 198].The associated permittivity of E. coli is εm = 2.5− 3.0 and S. cerevisae εm = 3.2− 4.2. Theseare in excellent agreement with the experimentally measured results of E. coli and S. cerevisaecell membranes [199].From Fig. 4.19 the resistance to electroporation from highest to lowest is: DphPC, E. coli,and S. cerevisae. As expected the resistance to electroporation increases as the tether densityincreases. The difference in the resistance to electroporation of DphPC compared to that ofE. coli and S. cerevisae is a result of the phytanyl chain packing properties and the ether link-ing the phytanyl to the lipid headgroup. The DphPC and GDEP ether-bound lipids found inarchaebacterial membranes are known to have a lower diffusion coefficient compared to com-mon phospholipids found in prokaryotes and eukaryotic membranes [200]. It is suggested in[201] that the stability of the DphPC membrane is closely related to the slow conformationalmotion of the phytanyl chains. Therefore we conclude that the difference in the resistanceto electroporation between the DphPC, and E. coli and S. cerevisae membranes is a result ofthe unique dynamics of the ether-phytanyl group which may result from a hydrogen bondingdifference. In addition, using the atomistic-to-observable model the conformational motion ofthe phytanyl tail in the DphPC decreases as the tether density increases which results in thelarger tether density membrane having a higher resistance to the effects of electroporation. Asurprising observation is that the E. coli membrane is more resistant to electroporation com-pared to that of S. cerevisae even though the thickness of S. cerevisae is larger than that of E.coli. A possible mechanism for this difference is that pores in the E. coli membrane are primar-ily formed by the flip-flop of specific groups of phospholipids [161]. From molecular dynamicssimulations [161] it is reported that the resistance to electroporation is a result of the reducedmobility of lipopolysaccharides, which comprises approximately 50% of the E. coli membrane,such that phospholipids primarily stabilize the aqueous pores. In comparison the S. cerevisaemembrane primarily contains the phospholipids dipalmitoylphosphatidylcholine, dioleoylphos-phatidylcholine, palmitoyloleoylphosphatidylethanolamine, palmitoyloleoylphosphatidylamine,and palmitoyloleoylphosphatidylserine with cholesterol [198]. Since lipopolysaccharides are notpresent in the S. cerevisae membrane, this reduces the energy for lipid flip-flop and thereforedecreases the resistance to electroporation compared to the E. coli membranes.874.4. Numerical and Experimental Resultsa bc de f10% DphPC 1% DphPC10% E. coli 1% E. coli10% S. cerevisiae 1% S. cerevisiaeFigure 4.19: The measured and predicted current response of the 1% and 10% tether densityDphPC, S. cerevisiae, and E. coli membranes. The excitation potential Vs is defined by a linearramp of 100 V/s for 5 ms followed by a -100 V/s for 5 ms. Cell 1 and Cell 2 denote the flowcell number of the CED in which the measurement was made. All predictions are computedusing equation (4.14) with the parameters defined in Table 4.3.4.4.9 Experimental Measurements and Model ParametersAll experimental measurements were conducted at 20oC in a phosphate buffered solution witha pH of 7.2, and a 0.15 M saline solution composed of Na+, K+, and Cl−. At this temperaturethe tethered membrane is in the liquid phase. A pH of 7.2 was selected to match that typicallyfound in the cellular cytosol of real cells. The quality of the tethered membrane is measuredcontinuously using an SDx tethered membranes tethaPodTM swept frequency impedance readeroperating at frequencies of 1000, 500, 200,100,40,20,10,5,2,1,0.5,0.1 Hz and an excitation po-tential of 20 mV (SDx Tethered Membranes, Roseville, Sydney). Custom drive potentials areproduced and the resulting current recorded using an eDAQTM ER466 potentiostat (eDAQ,Doig, Denistone East) and a SDx tethered membrane tethaPlateTM adaptor to connect to theassembled electrode and cartridge.884.4. Numerical and Experimental ResultsCGMD simulations were performed using GROMACS. PDEs were solved using the com-mercially available finite element solver COMSOL Multiphysics 4.3a. The coupled nonlinearFODEs (equation (4.14) and (4.12)) are solved using the Gru¨nwald-Letnikov method [202] withthe pore radii updated using the method presented in [3]. The governing equations (4.3) withboundary conditions (4.4) to (4.7), are solved numerically with the commercially available finiteelement solver COMSOL 4.3a (Comsol Multiphysics, Burlington, MA). To solve the GPNP andPNP models the COMSOL modules Transport of Diluted Species and Electrostatics are utilized;and to solve the EM model the modules Nernst-Planck and Electrostatics are utilized. The sim-ulation domain is meshed with approximately 270,000 triangular elements constructed usingan advancing front meshing algorithm. The GPNP and PNP are numerically solved using themultifrontal massively parallel sparse direct solver [107] with a variable-order variable-step-sizebackward differential formula [108]. Equation (4.8) is used to compute the pore conductancewith the integration done in the region defined in Fig. 4.3. The conductance is computed fora finite number of equally spaced radii between 0.5-10 nm with a step-size of 0.25 nm. Thesteady-state conductance Gp in (4.8) is estimated when the percentage change in conductancebetween successive steps (i.e. |(Gp(ti+1)−Gp(ti))/Gp(ti)|) is less then 1%. The total electricalenergy required to form the pore Wes, (4.9), is computed using the results from the conductancecomputation.To compute hm the particle density of the lipid headgroups is used. Fig. 4.20 and Fig. 4.21provides the normalized particle density for the CGMD beads W, PO4, NC3, OH, COC, andthe first C1 of the DphPC, and GDPE lipids for the 0% and 25% tethered membranes. Recallfrom section 4.3.1 that COC is associated with the ether linker in the DphPC and GDPE lipids.In Fig. 4.20 and Fig. 4.21 the distinct peaks in the OH and PO4 headgroups beads of GDPEand DphPC indicated the membrane is intact with negligible defects. As expected, the peak inthe particle density in Fig. 4.21 for water at 2.4 nm occurs between the OH bead of the spacerand the headgroup of the membrane.0 2 4 6 8 10 12 14 1600.20.40.60.81Position [nm]NormalizedDensityρ/ρ o  WPO4NC3OHCOCC1Figure 4.20: Normalized particle density computed from CGMD for the 0% DphPC tetheredmembrane.894.4. Numerical and Experimental Results0 2 4 6 8 10 12 14 1600.20.40.60.81Position [nm]NormalizedDensityρ/ρ o  WPO4NC3OHCOCC1Figure 4.21: Normalized particle density computed from CGMD for the 25% DphPC tetheredmembrane.To capture the essential dynamics needed for computing φ and J i needed for the evalua-tion of the pore conductance Gp and the electrical energy required to form a pore Wes, thefollowing assumptions are made. First, the diffusion coefficients in Di are assumed to be equal(i.e. isotropic diffusion). Second, the diffusion of Na+, K+, and Cl− is assumed to changeproportionally with how the diffusion of water varies. The third assumption is that the freeenergy of Na+, K+, and Cl− are all constant such that ∇F iw = 0. With these assumptions,the material parameters in Table 4.2 are used to solve equation (4.3) in the simulation domaindefined in Fig. 4.3 with diffusion coefficient is given by:Di(x) =Dir if x ∈ ΩrDiw if x ∈ Ωw.(4.18)The dielectric permittivity in equation (4.3) is spatially dependent as defined below:ε(x) =εw if x ∈ Ωr ∪ Ωwεm if x ∈ Ωm.(4.19)In Table 4.2, the concentrations match those used in the experimental measurements of theCED. The selection of effective ion size (i.e. solvated ionic radius) is based on the mobility mea-surements reported in [203]. The diffusion coefficients of the ions and electrical permittivitiesof water and biological membrane are provided in [204] and from the CGMD simulations. Thegeometric parameters hr and hm are selected to match the experimentally measured resultsobtained from neutron-reflectometry measurements of similar tethered membranes reported in[8] and the CGMD simulation results. The parameters Go, Cm, Cdl, and Re in Table 4.3 areestimated using a single impedance measurement for each tethered membrane. The electropo-ration parameters C,D, rm are obtained from [46, 53, 174, 175, 205]. The parameters σ andγ are computed from the CGMD simulations. Since α and q are not dependent on the tetherdensity, only a single current measurement was used to estimate these parameters, and foundto be consistent with those reported in [205].904.4. Numerical and Experimental ResultsThe pore density ρo can be estimated usingρo =GoAmGp(rm)(4.20)with Am = 2.1 mm2 the area of the membrane, and Go and Gp(rm) defined in Table 4.3.Table 4.2: Parameter for Gp and Wes PredictionsSymbol Definition ValuecNa|t=0 Initial Na+ concentration 321.45 mol/m3cK|t=0 Initial K+ concentration 13.39 mol/m3cCl|t=0 Initial Cl− concentration 334.84 mol/m3aNa Na+ effective ion size 4 A˚aK K+ effective ion size 5 A˚aCl Cl− effective ion size 4 A˚DNaw Na+ diffusion coefficient in Ωw 1.33× 10−9 m2/sDKw K+ diffusion coefficient in Ωw 1.96× 10−9 m2/sDClw Cl− diffusion coefficient in Ωw 2.07× 10−9 m2/sDNar Na+ diffusion coefficient in Ωr 0.92× 10−9 m2/sDKr K+ diffusion coefficient in Ωr 1.37× 10−9 m2/sDClr Cl− diffusion coefficient in Ωr 1.43× 10−9 m2/sεw Electrolyte electrical permittivity 7.083× 10−10 F/mεm Membrane electrical permittivity 2.65× 10−11 F/mF Faraday constant 9.6485× 104 C/molCs Stern layer capacitance 1 pFkB Boltzmann constant 1.3806488× 10−23 J/KT Temperature 300 Kφe Electrode potential 100-500 mVφec Counter electrode potential 0 mVlr Tether reservoir length 400 nmhr Tether reservoir height 4 nmhm Membrane thickness 3.5 nmhe Electrolyte height 60 nmNote that the unit nF∗ in Table 4.3 is defined as the SI unit s(p+3)/pA2/pm2/pkg1/p.914.5. Closing RemarksTable 4.3: Parameters for CED Current PredictionsSymbol Definition Valueγ Edge energy 1.2×10−11 J/mσ Surface tension 15×10−3 J/m2α Creation rate coefficient 1×109 s−1q q = (rm/r∗)2 2.46C Steric repulsion constant 9.67×10−15 J1/4 mD Radial diffusion coefficient 1×10−14 m2/srm Equilibrium pore radius 0.8 nmGp(rm) Equilibrium pore conductance 1.56 nSRe Electrolyte resistance 100-800 ΩSpacer Surface:p Fractional order parameter 0.83Cdl Double-layer capacitance 230 nF∗DphPC Membrane Tether Density: 1% 10% 100%G0 Initial membrane conductance 1.00 µS 0.66 µS 0.33 µSCm Membrane capacitance 16.0-17.5 nF 12.5-16.0 nF 12.4 nFp Fractional order parameter 0.90-0.95 0.90-0.95 0.93Cdl Double-layer capacitance 100-180 nF∗ 100-180 nF∗ 120-180 nF∗Vep Voltage of electroporation 350-415 mV 480-560 mV 650 mVKt Spring constant 0 N/m 0 N/m 20 mN/mS. cerevisiae Membrane Tether Density: 1% 10%G0 Initial membrane conductance 5.00 µS 1.11-1.66 µSCm Membrane capacitance 16.0-18.0 nF 14.0 nFp Fractional order parameter 0.90 0.90-0.92Cdl Double-layer capacitance 180 nF∗ 180 nF∗Vep Voltage of electroporation 330-350 mV 410-430 mVKt Spring constant 0 N/m 0 N/mE. coli Membrane Tether Density: 1% 10%G0 Initial membrane conductance 2.00-1.00 µS 0.66 µSCm Membrane capacitance 14.0 nF 15.0-17.0 nFp Fractional order parameter 0.90-0.91 0.90-0.91Cdl Double-layer capacitance 180 nF∗ 180 nF∗Vep Voltage of electroporation 360-380 mV 400-450 mVKt Spring constant 0 N/m 0 N/m4.5 Closing RemarksThis chapter has described the construction, measurement, and modeling of a novel cell-basedbioelectronic interface for controlled measurement of electroporation. The novel atomistic-to-observable (molecules to experimental measurement) model was derived to link the resultsfrom atomistic simulations with the experimentally measured current response from the device.Using the device and model several key features were presented including the dynamics ofwater, the structure and biomechanics of the device, conductance of aqueous pores, and theexistence of diffusion limited processes at the electrode surface. Despite the complexity of thedynamics in the controllable electroporation measurement device, it is shown that the threelevel model can accurately predict the macroscopic experimental measurements with differenttethering densities, membrane compositions, and drive voltages.925Electrophysiological ResponsePlatform5.1 IntroductionThis chapter focuses on the construction, predictive models, and experimental measurementsusing the electrophysiological response platform (ERP) of use for measuring the response ofembedded ion channels and cells grown on the tethered membrane surface. The ERP is designedto measure the electrophysiological response of cell and embedded ion channels in controllablemembrane and electrolyte environments which allows the platform to be used for drug screeningand diagnosing diseases in which ion-channel functionality is disrupted (i.e. channelopathies).Two applications of the ERP are presented in this chapter. The first is the measurement of theelectrophysiological response of the voltage-gated NaChBac ion channel embedded in the ERP.The second is the electrophysiological measurement of skeletal myoblasts grown on the surfaceof the ERP.5.1.1 Electrophysiological Response of the NaChBac Ion ChannelVoltage-gated ion channels are specialized proteins that only allow the passage of particularions at a rate determined by the electric potential gradient across the membrane (i.e. the trans-membrane potential). Voltage-gated channels initiate action potentials in nerve, muscle andother excitable cells and are vital for transcellular communication. In this chapter the ERP isapplied to the analysis of the prokaryotic sodium channel NaChBac, from Bacillus halodurans.The NaChBac channel was first reported in 2001 [22], and is likely an evolutionary ancestor ofthe larger four domain sodium channels in eukaryotes [23, 24]. The gating dynamics of NaCh-Bac have been studied using patch-clamping [206]; however, given the slow kinetics of NaChBacit was not possible to detect the gating dynamics at room temperature. In [206] the patch-clamping measurements were therefore performed at an elevated temperature of 28◦C. Theresults in [206] suggest that NaChBac has several closed states with voltage-dependent transi-tions. Using the ERP it is possible to measure the electrophysiological response of NaChBac atroom temperature in an engineered tethered membrane that mimics the response of biological935.2. Formation and Operationmembranes. The advantage of using the tethered membrane compared to that of black lipidmembranes for the study of NaChBac gating dynamics is its stability and robustness. This al-lows the effects of electroporation to be accounted for while measuring the electrophysiologicalresponse of NaChBac in the tethered membrane.5.1.2 Electrophysiological Response of CellsThe electrophysiological measurement of cells allows for the diagnosis of channelopathic dis-eases such as cystic fibrosis and Bartter’s syndrome. Consider that in cystic fibrosis, the cysticfibrosis transmembrane conducting regulator protein blocks the flow of chloride and thiocyanateions, which cause a decrease in the cell membrane conductance. The ERP utilizes an engineeredtethered membrane for the non-invasive electrophysiological measurement of cells which can beused to detect such a change in conductance. Typically an adhesion protein, such as Glycocalyx,is used ensure there is sufficient adhesion of the cell to the sensing surface [68]. Using a suit-ably designed tethered membrane, cells can reach a sufficient coverage and adhesion to allow fortheir electrophysiological measurement. It was determined that a 100% tethered archaebacterialbased monolayer provides a suitable membrane to promote cell growth for electrophysiologicalmeasurement. Remarkably this allows the response to be measured using a non-invasive tech-nique in the proximity of a synthetic membrane that mimics the electrophysiological responseof biological membranes.5.1.3 Main Results and OrganizationThe main contribution of this chapter is to illustrate how experimental measurements fromtethered membranes and dynamic model can be used to estimate the electrophysiological re-sponse of embedded ion channels, and cells. Sec. 5.2 provides the formation and operation ofthe ERP for embedding ion channels into the tethered membrane, and for the growth of cells onthe tethered membrane surface. Fractional order macroscopic models of the ERP are providedin Sec. 5.3. Using experimental measurements from the ERP and dynamic model allow theelectrophysiological response of ion channels and cells to be estimated. In Sec. 5.4, the ERPand dynamic model are utilized to measure the electrophysiological response of the NaChBacion channel, and Skeletal myoblasts which are attractive donor cells for cardiomyoplasty usedto regenerate damaged myocardium tissue produced from acute myocardial infarction [25, 26].Closing remarks are provided in Sec. 5.5.5.2 Formation and OperationThe ERP can be used to measure the electrophysiological response of a) ion channels embeddedin the tethered membrane of the ERP, and b) cells grown on the tethered membrane surface of945.2. Formation and Operationthe ERP.The formation of the ERP for ion channels which spontaneously insert into the membraneis identical to that presented in Sec. 3.2. For ion channels that do not spontaneously insert thechannels must be inserted when the tethered membrane is being formed. For the analysis ofthe voltage-gated NaChBac ion channel the ERP contains a palmitoleic phytanyl phosphatidyl-choline tethered membrane with a 10% tether density. The following protocol is used to acquireand insert the NaChBac channels into the ERP for measurement2.• The first step is to begin with cells that contain NaChbac with a polyhistidine-tag. Inthis analysis the tagged NaChBac are obtained from Bacillus halodurans, a Gram-positivebacterium found in soil.• The second step is to fragment the cells to extract the polyhistidine-tagged NaChBac pro-teins. This can either be done using a French press, or mortar and Pestle Fragmentationprotocol.• The third step involves anchoring the cell fragments in a mini-column. The anchored cellsare then rinsed with imidazole to release the tagged NaChBac proteins.• The fourth step is to rinse the released NaChBac proteins with a detergent. Several deter-gents can be used, however it is recommended to use a detergent with a high aggregationnumber such as Brij 58, CYMAL-5, TWEEN-20, Triton X-100, or DDM. For this analysisthe CYMAL-5 detergent micelles are used to produce a solution containing 100 nM ofNaChBac.• Add the solution containing 100 nM of NaChBac to the mobile lipid solution and proceedwith the formation of the membrane as outlined in Sec. 3.2.For the electrophysiological measurement of cells, the tethered membrane is required tohave a 100% tethering density to ensure the tethered membrane remains intact during cellgrowth and measurement. The formation of the 100% tethered membrane is identical to thatpresented in Sec. 3.2. Progenitor cells are used to produce skeletal myoblast cells. To inducedifferentiation, we exposed skeletal myoblasts to a range of growth factor and small moleculecomponents currently commercial-in-confidence to Genea BiocellsTM. After thawing Geneaskeletal muscle cells, they were re-suspended in Genea Skeletal Muscle Thawing Solution andplated onto eight 600 µL wells in a cartridge containing sixteen individually read wells. Eachcontained a synthetic archaebacterial tethered membrane, refer to Fig. 3.1. The membrane isconstructed with a radial dimension of 3 mm2 and was bounded by a polypropylene chambercompressed to form a hermetic seal onto the membrane surface. The passaging buffer wasimmediately replaced with 450 µL of PBS and the sixteen chamber cartridge transferred to2www.sdxtetheredmembranes.com/applications955.3. Dynamic Modela 37oC, 75% RH incubation cabinet. During incubation cell coverage was monitored usingimpedance measurements from an excitation potential of 30 mV.5.3 Dynamic ModelIn this section two fractional order macroscopic models of the ERP are provided for measuringthe electrophysiological response of embedded ion channels, and cells grown on the surface ofthe tethered membrane.5.3.1 Embedded Ion ChannelsTo measure the electrophysiological response of embedded ion channels using experimentalmeasurements from the ERP requires the dynamics of the tethered membrane be fully defined.The dynamics of the tethered membrane are modelled using the fractional order model (4.14)and (4.12). Given that ion channels are present in the tethered membrane they also contributeto the conductance dynamics. To account for the conductance of ion channels, denoted byGc, in the tethered membrane the parameter Gc is included in parallel with the aqueous poreconductance in the tethered membrane Gm. The complete fractional order macroscopic modelis provided in Fig. 5.1. To estimate the embedded ensemble ion channel conductance Gc,Cbdl+− VsGm Cm+−VmGcReCtdlI(t)Figure 5.1: Fractional order macroscopic model of the ERP for measuring embedded ion channelconductance Gc. Vm is the membrane potential, Vs the applied potential, and I(t) the currentresponse. The circuit parameters are described in Sec. 5.3.the experimentally measured current I(t) is compared with the numerically predicted currentfrom macroscopic fractional order model in Fig. 5.1. When agreement between the numericallypredicted and measured current is reached, then Gc denotes the electrophysiological responseof the embedded ion channels.965.4. Numerical and Experimental Results5.3.2 Cell ResponseSimilar to the measurement of embedded ion channels, the estimation of the electrophysio-logical response of cells using experimental measurements from the ERP requires a dynamicmodel. In this section a fractional order model is provided that can be used to estimate theelectrophysiological response of cells using experimental measurements from the ERP.The cellular suspension is modelled by considering a distributed parallel network of cellswith uniform properties (i.e. each cell has identical membrane capacitance Cc [207], cytoplasmresistance Rc [208, 209], and the conductive properties of each patch of cell are uniform). Giventhe cells have uniform characteristics, the entire cell suspension can be represented by an equiv-alent circuit as illustrated in Fig. 5.2. Note that RL models the leakage resistance caused bycurrent flowing directly from the tethered membrane surface to the electrolyte solution [210].The circuit parameters Ctdl, Re, Gm, Cm, and Cbdl are defined in (4.14). To account for the dif-ferent polarizations of the top and bottom of the cell membrane we have defined the parametersCtm, Gtl, and Gtm for the capacitance, leakage conductance, and variable conductance of thetop surface of the cell adjacent to the bulk electrolyte, and Cbm, Gbl, and Gbm for the bottomsurface of the cell adjacent to the tethered membrane surface. Rcm is the total cytoplasmicresistance of the cellular suspension. In Fig. 5.2, Vr denotes the resting potential of the cellmembrane maintained by leakage ion channels.The membrane capacitances Ctm and Cdm in Fig. 5.2 are directly proportional to the frac-tional surface area of the top and bottom cell membranes. Denote At as the surface area of thetop of the cell, and Ab as the surface area of the bottom of the cell then the total cell membranecapacitance Ctot = AtCc +AbCc is related to Ctm and Cbm as follows:Ctm = AfCtot Cbm = (1−Af )Ctot,Af = At/(At +Ab). (5.1)Eq.(5.1) allows us to account for the geometry of the cells in a meaningful way. Note that for acompletely symmetric cell Af = 0.5; however, it is well known that cells typically do not formsymmetric structures on a flat surface such that Af > 0.5 [211].To estimate the membrane conductance Gbm and Gtm, the experimentally measured currentI(t) is compared with the numerically predicted current from the lumped circuit in Fig. 5.2.When agreement between the numerically predicted and measured current is reached, then Gbmand Gtm denote the electrophysiological response of the cell membrane.5.4 Numerical and Experimental ResultsIn this section experimental measurements from the ERP and fractional order macroscopicmodels (Fig. 5.1) for embedded ion channels, and (Fig. 5.2) for cells, are used to estimate the975.4. Numerical and Experimental ResultsCellSuspensionTetheredMembraneGtmGlt+−VrCtm+−VtmRcmGbm +−VrGlbCbm−+VbmGm Cm−+VmRLCbdlRe Ctdl+−VsI(t)Figure 5.2: Lumped circuit model of the engineered tethered membrane with cells. Vs is thedriving voltage, Vtm is the potential across the cell membrane adjacent to the bulk electrolyte,Vbm is the the potential across the cell membrane adjacent to the tethered membrane surface,and Vm is the tethered membrane potential. The circuit parameters are described in Sec. 5.3.electrophysiological response of the NaChBac ion channel, and skeletal myoblasts.5.4.1 Electrophysiological Response of NaChBac Ion ChannelsTo study the electrophysiological response of the embedded voltage-gated NaChBac ion channelthe excitation potential, given in Fig. 5.3(a), is applied to the ERP and the associated currentresponse recorded. The excitation potential in Fig. 5.3(a) is composed of increasing and de-creasing voltage ramps with different slopes. The experimentally measured and numericallypredicted current response is provided in Fig. 5.3(b). As seen the experimentally measuredand numerically predicted current response are in excellent agreement suggesting the estimatedconductance Gc is a valid representation of the ensemble conductance of the NaChBac chan-nels. From Fig. 5.3(c) and Fig. 5.3(d), for positive transmembrane potentials the estimatedconductance of NaChBac channels Gc increases slightly, however there is a dramatic increase inconduction for negative transmembrane potentials. Like most voltage-gated channels, NaChBacinactivates for positive transmembrane potentials [206]. Why is there an increase in conductancefor a positive transmembrane potential? A possible reason is there is a population difference inthe orientation of the embedded NaChBac ion channels in the tethered membrane. Given the985.4. Numerical and Experimental Resultsconductance of a NaChBac channel is approximately 12 pS [22] and assuming the NaChBacis in a closed state for Vm > 0, it is possible to estimate the total population of NaChBacchannels in each direction from the two peaks occurring at 1.7 µs and 2.2 µs in Fig. 5.3(d).At 1.7 µs the conductance is 23 µS corresponding to 1.9 million conducting channels, and at2.2 µs the conductance is 170 µS corresponding to 14.1 million conducting channels. Given thetethered membrane has a surface area of 2.1 mm2, with 14.1 million conducting channels thiscorresponds to each channel occupying an area of approximately 150,000 nm2. From Fig. 5.3(d)notice that the contribution of electroporation is negligible compared to the conductance dy-namics resulting from the NaChBac channel. Note that if the population of NaChBac channelsdecreased then the effects of electroporation must be accounted for.0 5 10−600−400−2000200Time [ms]V s(t)[mV]0 5 10−20−15−10−50Time [ms]Current[µA]  ExperimentalPredicted0 5 10−200−1000100Time [ms]V m(t)[mV]0 5 10050100150Time [ms]G c,G m[µS]  Gc(t)Gm(t)a bc dFigure 5.3: Experimentally measured and numerically predicted electrophysiological response ofNaCHBac ion channels in the ERP. The numerical predictions are computed using the modelin Fig. 5.1 with the parameters defined in Table 4.3. Panel (a) is the drive potential Vs(t).(b) Experimentally measured and numerically predicted current response. (c) Predicted trans-membrane potential Vm(t). (d) Predicted total aqueous pore Gm(t) and NaChBac conductanceGc(t).5.4.2 Electrophysiological Response of CellsIn this section the ERP and macroscopic fractional order model (Fig. 5.2) is used to measurethe electrophysiological response of skeletal myoblasts.995.4. Numerical and Experimental ResultsTo measure the electrophysiological response of the skeletal myoblasts, a sawtooth voltageexcitation waveform was utilized with a slope of 250 V/s for 2 ms, and another with a slopof -250 V/s for a 2 ms. If the leakage current between cells was dominant, the measuredcurrent response I(t) in (Fig. 5.2) would be dominated by the current flowing through theleakage resistance RL, and the electrophysiological response of cells would be unobservable.Additionally, if the geometry of cells in the monolayer are symmetric then the current responsefrom the linearly increasing and decreasing drive potentials would be related by a sign difference.Fig. 5.4 provides the numerically computed and experimentally measured current response of theskeletal myoblasts to the two sawtooth voltage excitation waveforms. Referring to Fig. 5.4(a)and Fig. 5.4(c), the current response is asymmetric. Therefore we can conclude that RL isnot dominant, and Af > 0.5 (5.1). Using the measured current response in Fig. 5.4(a) andFig. 5.4(c) with the dynamic model (Fig. 5.2), we estimate that Af = 0.75. The estimated Afis consistent with values computed for neurons on electrode surfaces [211].Consider the charging dynamics of the membranes resulting from the linearly increasing po-tential Vs(t) producing the current response provided in Fig. 5.4(a). For the linearly increasingpotential Vs(t) the top membrane adjacent to the bulk electrolyte experiences depolarization asthe transmembrane voltage Vtm increases above the membrane rest potential Vr=-70 mV pro-moting the activation of voltage-gated ion channels on the top membrane surface. As the thetop membrane depolarizes, the bottom membrane adjacent to the tethered membrane is under-going hyperpolarization as the transmembrane voltage Vbm decreases below -70 mV inhibitingany voltage-gated ion channels from activating. Therefore, for a linearly increasing potentialwe expect Gtm to vary by contributions from leakage ion channels (i.e. non-selective ion chan-nels which are always open) and electroporation effects while Gbm includes effects caused byleakage ion channels, electroporation, and voltage-gated ion channels. For the current responsein Fig. 5.4(a) the transmembrane potential Vtm is in the range of -70 mV to 50 mV, and Vbm isin the range of - 70 mV to -160 mV. For these transmembrane potentials, there was no voltage-gated ion channel dynamics detected–that is, if the dynamic model in (Fig. 5.2) only includesthe effects of electroporation for the tethered membrane, top cell membrane, and bottom cellmembrane surfaces then the experimental results are in agreement with the dynamic model.The associated conductance for the three membranes resulting from the excitation potentialproducing the results in Fig. 5.4(a) are provided in Fig. 5.4(b). As expected, since the magnitudeof Vbm is larger then Vtm, the effects of electroporation are more pronounced on Gbm comparedto that of Gtm. For a linearly decreasing excitation potential Vs(t) the bottom membrane de-polarizes which activated voltage-gated ion channels–that is, the results of the dynamic model(Fig. 5.2) are not in agreement with the experimentally measured results if only electroporationis considered. This effect is also evident as the current response in Fig. 5.4(a) and Fig. 5.4(c)are not related by a sign change even though the excitation potential Vs(t) is related by a signchange. Therefore, given a series of conductance measurements of the bottom membrane, an1005.5. Closing Remarksexperimenter could perform parameter estimation of an ensemble of ion-channel gating models(i.e. coefficient and exponents in ODEs of Hodgkin-Huxley type models or transition probabil-ities between states in discrete-state Markov models) [212]. Therefore the engineered tetheredmembrane with adhered cells can be used for validating gating models of interacting and non-interacting ion channels and for the measurement of the electrophysiological response of cells.0 1 2 3 4−1−0.8−0.6−0.4−0.200.2Time [ms]Current I(t) [mA]  PredictedExperimental250 V/s0 1 2 3 412345Time [ms]Conductance [mS/cm2 ]  Gm(t)Gbm(t)Gtm(t)250 V/s0 1 2 3 4−0.200.20.40.60.81Time [ms]Current I(t) [mA]  PredictedExperimental −250 V/s0 1 2 3 412345Time [ms]Conductance [mS/cm2 ]  Gm(t)Gbm(t)Gtm(t)−250 V/sCurrent[µA]Conductance[µS/mm2] m t)b (t)t (t)//ime [ms] Time [ms]Current[µA]Conductance[µS/mm2] m t)b (t)t (t)-250 /-250 /ime [ms] Time [ms]a bc dFigure 5.4: Panel (a) presents the experimentally measured and numerically predicted currentresponse I(t) for a sawtooth pulse drive potential Vs(t) with a slope of 250 V/s for 2 ms. (b) isthe computed membrane conductance Gm, Gbm the cells membrane conductance adjacent to thetethered membrane, and Gtm the conductance of the cell membrane facing the bulk electrolytesolution computed using the current response in (a). (c) presents the experimentally measuredand numerically predicted current response I(t) for a sawtooth pulse drive potential Vs(t) witha slope of -250 V/s for 2 ms. (d) is the computed membrane conductance Gm, Gbm the cellsmembrane conductance adjacent to the tethered membrane, and Gtm the conductance of thecell membrane facing the bulk electrolyte solution computed using the current response in (c).All predictions are computed using (4.14) and (4.12) with the parameters defined in Table 4.3.5.5 Closing RemarksIn this chapter the formation and operation, dynamic models, and experimental measurementsusing the electrophysiological response platform (ERP) were presented. Experimental measure-ments from the ERP and dynamic model allow the electrophysiological response of embedded1015.5. Closing Remarksion channels and cells to be estimated in controllable electrolyte and tethered membrane envi-ronments. The fractional order macroscopic model accounts for the diffusion limited processesat the electrode surface of the ERP and the effects caused by the process of electroporation.Two applications of the ERP and dynamic model were provided. In the first the ERP anddynamic model were utilized to study the electrophysiological response of the voltage-gatedNaChBac ion channel in a palmitoleic phytanyl phosphatidylcholine tethered membrane. Thesecond applied the ERP and dynamic model to study the electrophysiological response of skele-tal myoblasts grown on the surface of the tethered membrane. In future applications the ERPand dynamic model can be used to validate ion-channel gating models (i.e. coefficient andexponents in ODEs of Hodgkin-Huxley type models or transition probabilities between statesin discrete-state Markov models) [212], and for the discovery of novel channel blocking drugs.1026ConclusionsThe unifying theme of this thesis was to present the construction and predictive models of fourtethered membrane measurement platforms: (i) the Ion Channel Switch (ICS) biosensor fordetecting the presence of analyte molecules in a fluid chamber, (ii) the Pore Formation Mea-surement Platform (PFMP) for detecting the presence of pore forming proteins and peptides,(iii) the Controlled Electroporation Measurement Device (CED) that provides reliable measure-ments of the electroporation phenomenon, and (iv) the Electrophysiological Response Platform(ERP) to measure the response of ion channels and cells to an electrical stimulus. Common toall four measurement platforms is that: (a) they are comprised of a self-assembled engineeredtethered membrane that is formed via a rapid solvent exchange technique allowing the platformto have a lifetime of several months, (b) they all use an inert gold bioelectronic interface, (c)their dynamics can be modeled using a fractional order model coupled with continuum modelsand coarse-grained molecular dynamics, and (d) the sensing mechanism of each relies on mea-suring the changes in tethered membrane conductance. Each of the Chapters 2 to 5 presentthe formation, dynamic models, and experimental measurements for each of the four tetheredmembrane platforms. The dynamic models are composed of three levels of abstraction: coarse-grained molecular dynamics, continuum models for electrodiffusive flow and chemical reactions,and macroscopic fractional order model for electroporation and the diffusion limited processpresent at the electrode surface of the tethered membranes. This chapter completes the studyby summarizing the main results and their implications.The main results presented in this thesis are the development of dynamic models whichallow experimental measurements from the four tethered membrane platforms to be used toestimate important biological parameters. These results are summarized as follows:1. In Chapter 2 a fractional order model, Poisson-Nernst-Planck, and surface reaction-diffusion equations are coupled to estimate the tethered membrane conductance of theICS biosensor for a given input analyte concentration. It was established that diffusion-limited charge transfer, reaction limited charge transfer, and ionic adsorption dynamicsare present at the gold electrode of the ICS which can be accounted for using a fractionalorder model. Using experimental measurements from the ICS and dynamic model theconcentration of streptavidin, thyroid stimulating hormone (TSH), ferritin, and humanchorionic gonadotropin (hCG) was estimated. The estimated concentration and knowninput concentration are in excellent agreement.1036. Conclusions2. In Chapter 3 a fractional order model, a generalized reaction-diffusion model, and coarse-grained molecular dynamics model were constructed and applied to estimate the tetheredmembrane conductance of the PFMP to estimate the pore formation dynamics of PGLa.Using the CGMD model, the dynamics of surface binding, translocation of surface boundto transmembrane bound, and the oligomerization of transmembrane bound PGLa wasstudied. The dynamic model was validate using experimental measurements of the poreforming toxin α-Hemolysin produced by Staphylococcus aureus. Using experimental mea-surements from the PFMP and dynamic model the reaction pathway of PGLa pore for-mation dynamics in charged and uncharged membrane surfaces that mimic prokaryotic,eukaryotic, and archaebacterial membranes were presented.3. In Chapter 4 a fractional order model, a generalized Poisson-Nernst-Planck model, andcoarse-grained molecular dynamics model were constructed and applied to: compute thediffusion tensor of water, membrane thickness, aqueous pore density, conductance of aque-ous pores, energy for lipid flip-flop, and estimate the tethered membrane conductance ofthe CED to an applied voltage potential. Using the CGMD model, key findings of thebioelectronic interface were presented including: Fick’s law of diffusion applies at thesurface of the membrane and in the tethering reservoir, the diffusion tensor of water isspatially dependent, membrane thickness, lipid diffusion, pore density, the free energy oflipid flip-flop, and the effective dielectric permittivity of the membrane are dependent onthe tether density. The results of the GPNP model illustrated that the conductance ofaqueous pores in the tethered membrane are linearly proportional to the radius of thepore and the not the radius squared as typically assumed [21]. This unique feature is aresult of the proximity of the membrane to the gold surface. The dynamic model wasvalidate using experimental measurements of tethered DphPC, E. coli, S. cerevisae lipidmembranes for different tethering densities.4. In Chapter 5 fractional order models to estimate the electrophysiological response ofembedded ion channels and cells grown on the surface of the ERP are presented. Thedynamic model was used to estimate the electrophysiological response of the voltage-gated NaChBac channel, and the response of skeletal myoblasts grown on the tetheredmembrane surface of the ERP.Though the dynamic models are constructed to estimate important biological parameters fromexperimental measurements from the ICS, PFMP, CED, and ERP, the models provide valuabletools which are of use for any sensing device constructed with an inert metal bioelectronic inter-face and/or a tethered membrane. These include transport-binding experiments, for examplethe analysis of thin film deposition processes.There is much work left to be done in modeling and utilizing experimental measurementsfrom sensing devices constructed from biological membranes. In the following several directions1046. Conclusionsfor potential future research along the lines of this thesis are provided:1. The dynamic model developed for the ICS can be applied for design purposes to select thebinding site concentration, flow rate, and geometric parameters of the ICS flow chamberfor the detection of specific analyte molecules of interest. For charged analyte moleculesthe design of excitation potential to enhance the detection ability of the ICS can also beperformed using the dynamic model.2. Each of the tethered membrane platforms presented in Chapters 2 to 5 contain an inertgold bioelectronic interface which is modeled using a fractional order macroscopic model.What is the structure (i.e. density profile of ions and water) near the bioelectronic in-terface? In [213] the equivalence between the anomalous diffusion Poisson-Nernst-Planckequation and the fractional order model of the bioelectronic interface was established,however this does not provide a molecular level description of the bioelectronic interface.The study of the bioelectronic interface at the molecular level is a formidable task, referto citations in [103] for details. First-principle simulations of the metal-aqueous interfacecan only capture dynamics on the scale of a few picoseconds. Molecular dynamics how-ever, with force fields such as GolP-CHARMM [214] which capture the polarizability ofgold atoms, can reach the nanosecond timescale. Therefore molecular dynamics offers amethod which can be used to gain valuable insight into the structure of ions and waterin proximity to the bioelectronic interface in the tethered membrane platforms.3. In Chapter 5 the ERP and dynamic model were presented to study the electrophysi-ological response of embedded ion channels and cells. In standard patch-clamping theexperimentalist can directly control the transmembrane potential which governs the dy-namics of voltage-gated ion channels. In the ERP the experimentalist can control thedrive voltage but not the transmembrane potential directly–which is dependent on theelectrical double layer charging effects at the bioelectronic interface and the conductanceof the membrane. Notice that for a given ion channel gating model, the fractional or-der macroscopic model presented in Chapter 5 can be used to predict the response ofthe ERP. Using a fractional-order controller [215], the experimentalist could set a desiredtransmembrane potential to validate the ion channel gating model that would be obtainedfrom standard patch-clamping techniques. 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