Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Computational study of a quantum mechanical based model of magnetoreception Garcia Chavez, Miguel Angel 2012

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2012_fall_garciachavez_miguelangel.pdf [ 407.8kB ]
Metadata
JSON: 24-1.0072778.json
JSON-LD: 24-1.0072778-ld.json
RDF/XML (Pretty): 24-1.0072778-rdf.xml
RDF/JSON: 24-1.0072778-rdf.json
Turtle: 24-1.0072778-turtle.txt
N-Triples: 24-1.0072778-rdf-ntriples.txt
Original Record: 24-1.0072778-source.json
Full Text
24-1.0072778-fulltext.txt
Citation
24-1.0072778.ris

Full Text

Computational Study of a Quantum Mechanical Based Model of Magnetoreception  by Miguel Angel Garcia Chavez BSc. Physics, University of the Americas, Puebla, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Master of Science in THE FACULTY OF GRADUATE STUDIES (Physics)  The University Of British Columbia (Vancouver) April 2012 c Miguel Angel Garcia Chavez, 2012  Abstract In this work we study a quantum mechanical model of magnetoreception in birds based on a pair of electrons undergoing coherent evolution. The magnetoreception arises because there is a different outcome depending on whether the electrons form a singlet or a triplet state, and a magnetic field can influence this configuration. We perform a variety of computational simulations on a simplified model of the above mechanism, with the purpose to study the effect of changing the different variables in which the process depends, ultimately to determine the plausibility of the mechanism.  ii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  2  Magnetoreception Mechanism . . . . . . . . . . . . . . . . . . . . .  3  3  Results and Discussion I: Coherent Evolution . . . . . . . . . . . . .  9  4  Results and Discussion 2: Triplet Yield . . . . . . . . . . . . . . . .  33  5  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  47  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  50  iii  List of Figures Figure 3.1  Single Spin one half, External field of one Gauss, hyperfine coupling one gauss . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.2  Single Spin one half, External field of one Gauss, hyperfine coupling ten gauss . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.3  22  Three Spin one half, External field of one Gauss, hyperfine coupling one gauss . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.9  21  Single Spin one half, External field of ten Gauss, hyperfine coupling ten gauss . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.8  20  Single Spin one half, External field of 100 Gauss, hyperfine coupling one gauss . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.7  19  Single Spin one half, External field of ten Gauss, hyperfine coupling one gauss . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.6  18  Single Spin one half, External field of one Gauss, hyperfine coupling 100 gauss . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.5  17  Single Spin one half, External field of one Gauss, hyperfine coupling 100 gauss . . . . . . . . . . . . . . . . . . . . . . .  Figure 3.4  16  23  Three Spin one half, External field of one Gauss, hyperfine coupling ten gauss . . . . . . . . . . . . . . . . . . . . . . .  24  Figure 3.10 Three Spin one half, External field of one Gauss, hyperfine coupling 100 gauss . . . . . . . . . . . . . . . . . . . . . . .  25  Figure 3.11 Three Spin one half, External field of one Gauss, hyperfine coupling 100 gauss . . . . . . . . . . . . . . . . . . . . . . .  iv  26  Figure 3.12 Triplet Probability vs. time, Three Spin one half, External field of ten Gauss, hyperfine coupling one gauss . . . . . . . . . .  27  Figure 3.13 Three Spin one half, External field of ten Gauss, hyperfine coupling ten gauss . . . . . . . . . . . . . . . . . . . . . . . . .  28  Figure 3.14 Three Spin one half, External field of 100 Gauss, hyperfine coupling one gauss . . . . . . . . . . . . . . . . . . . . . . .  29  Figure 3.15 Triplet Probability vs. time with decay constants ks=0 and kt=1 30 Figure 3.16 Triplet Probability vs. time with decay constants ks=1 and kt = 0 31 Figure 3.17 Triplet Probability vs. time with decay constants ks=1 and kt=1 32 Figure 4.1  Single Spin one half, hyperfine coupling one gauss . . . . . .  37  Figure 4.2  Single Spin one half, hyperfine coupling 100 gauss . . . . . .  38  Figure 4.3  Three Spin one half, hyperfine coupling one gauss . . . . . .  39  Figure 4.4  Three Spin one half, hyperfine coupling ten gauss . . . . . . .  40  Figure 4.5  Angle Dependence, Field one Gauss, Hyperfine Coupling one Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Figure 4.6  41  Angle Dependence, Field one Gauss, Hyperfine Coupling thirty Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  42  Angle Dependennce, Field one Gauss, Hyperfine Coupling sixty Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  43  Figure 4.8  Angle Dependence, Rate Constants kt = 2, ks = 0 . . . . . . .  44  Figure 4.9  Angle Dependence, Rate Constants kt = 1.5, ks = 0.5 . . . . .  45  Figure 4.10 Angle Dependence, Rate Constants, kt = 0.5, ks = 1.5 . . . . .  46  Figure 4.7  v  Acknowledgments I would like to thank the many people who make this possible. I want to thank my supervisor for this project, Dr. Steve Plotkin, for his valuable guidance and advice. I also want to thank all of my fellow members of our research group for their insightful discussions, specially Dr. Atanu Das for his help understanding the background material. I also want to thank NSERC for their financial support. And finally, I want to thank my family and friends for their encouragement and continous emotional support.  vi  Chapter 1  Introduction It is now well established that many species of birds use the Earth magnetic field to orient themselves during migration [30]. Wiltschko et al have published comprehensive experimental works on the experimental nature of the birds magnetoreception system [31] [21] [32]. One of the properties of the mechanism is that it cannot detect the polarity of the field [31], that is, it only detects the overall north-south direction, but not which way is north and which way is south. Another property is that birds are disoriented in the presence of an oscillating magnetic field of a magnitude much smaller than the Earth field [21]. There are two main hypothesis about the nature of this mechanism. One is that it is based on magnetite particles embedded in the bird physiology [12]. The other one is a quantum mechanical based model [20], which will be the focus of the present work. The main idea of the mechanism is to have two radicals each with an unpaired electron. If the electrons are close enough, the direction of their spins may become correlated. The Earth magnetic field can influence their relative orientation, and the ultimate fate of the radical pair will depend on this relative orientation. Hence we have a chemical reaction that can be in principle affected by the external magnetic field. This reaction can in turn trigger a biological pathway that would let the birds to effectively feel the magnetic field. It is hypothesized that this mechanism would involve the protein cryptochrome [6]. Cryptochrome is a large blue light photoreceptor protein that is known to mediate 1  a variety of light responses, like the circadian rhythm in some animals [2]. Cryptochrome binds the cofactor flavin adenine dinucleotide (FAD) [8], which is generally considered to be one of the radicals [19]. The other radical has been proposed to be a tryptophan [26], or an external superoxide molecule [12]. In the present work we present a critical analysis of the feasibility of the mechanism described above. For reasons that will be explained in the work, it was not possible to realistically model the system. Instead, we use a simplified model in which one of the radicals is represented either by a single spin one-half nuclei or a three spin one-half nuclei, and the other radical has zero nuclear spin. Since superoxide has zero nuclear spin (Oxygen nucleus is spinless, as are all nuclei with an even number of nucleons [14]) our model is closer to the FAD - superoxide interaction than to the FAD - Tryptophan one. Some calculations have been done before for the systems similar to ours [20] [26] [25], but the present work contains a more complete overview of the effect of varying the different quantities in which the model depends, namely the number of nuclear spins, the decay rate constants, the magnitude and direction of the external magnetic field, and the magnitude of the hyperfine coupling constants.  2  Chapter 2  Magnetoreception Mechanism The idea that a magnetic field could influence a chemical reaction by means of its effect on the singlet or triplet nature of a pair of spins was first proposed by Schulten [24]. That this mechanism could be behind the magnetic sensing of birds was proposed by Ritz [20]. In the following we are going to describe how this process can happen. This description mostly follows Ritz [20] and Timmel [27]. First, we require two radicals, each with one unpaired electron. We we call them radical A and B in this general discussion, although for our system of interest the identity of the radicals are FAD and superoxide or tryptophan. If radicals come close enough, the electrons may become entangled, which in this case means that their spins will form a single quantum state, which may be a singlet or a triplet state. Intuitively (although not exactly correct), in a triplet state the electron spins are parallel, and in a singlet state the electron spins are anti parallel. For this radical pair mechanism to work it is necessary that there is a different outcome for the singlet and triplet states. In this sense, the radical pair can be viewed as the transition state of a chemical reaction, where A and B are the reactants and there are two different types of products depending on the quantum state of the radical at the moment of the decay. From now on we will call ”triplet products” and ”singlet products” to the products formed from a triplet or singlet radical pair, respectively. We are going to work in the density matrix formalism of quantum mechanics [22], which allows for the description of statistical ensembles of states. In this 3  formalism, if a system has a probability pi of being in state ψi , the density matrix for the system is ∑i pi |ψi >< ψi | . Note that pi are classical probabilities, the probability that a system is in a specific quantum state, and are different from the quantum probabilities associated with the expansion of a state vector in its eigenfunctions. The density matrix satisfies the Liouville equation [27] i ∂ρ = − [ρ , H] h¯ ∂t  (2.1)  where ρ is the density matrix, H is the Hamiltonian and [A, B] = AB − BA. This Liouville equation is a generalization of the Schroedinger equation that is able to deal with ensembles, and it is the quantum version of the Liouville equation from classical mechanics. However, the Liouville equation, as the Schroedinger equation, only applies to closed systems. Our system is an ensemble of radical pairs, which is not a closed system, since it is expected that eventually all the radicals will decay into products. If we assume that the reaction rate is first order with respect to the concentration of radicals, then the amount of radicals will decay as a negative exponential in time [4], and the Liouville equation generalizes to: i ∂ρ ks kt = − [ρ , H] − {Qs , ρ (t)} − Qt , ρ (t) ∂t 2 2 h¯  (2.2)  Where ks and kt are the singlet and triplet rate constants, that is, the rate constants that determine how fast the radicals decay into products when they are in a singlet or a triplet state. Qs and Qt are the singlet and triplet projectors, respectively, which are operators that extract the portion of a system that is in a singlet or a triplet state. This projector operators are defined as [26]: Qs =  1 − S1 · S2 4  (2.3)  Qt =  3 + S1 · S2 4  (2.4)  Where S1 and S2 are the electron spin operators: Si = Six x + Siy y + Siz z 4  (2.5)  And Sx , Sy and Sz are the standard Pauli matrices. In this form, equation 2.2 is specific for the problem we are solving, since it assumes that the radicals can decay into products either from a singlet state (second term on the right hand side) or from a triplet state (third term on the RHS). This equation is sometimes called the stochastic Liouville equation [15]. To extract information from the density matrix once it has been obtained, we multiply it by the operator of interest and then take its trace. This is the standard procedure to determine the expectation value of an operator in the density matrix formulation. For example, to determine the probability that the system is in a triplet state at a specific time we do: T (t) = Tr[Qt ρ (t)]  (2.6)  Usually, we are interested in the total amount of triplet products. This quantity is called the triplet yield and is given by the integrals over all times of the triplet probability multiplied by the triplet rate constant. ΦT =  ∞ o  kT T (t)dt  (2.7)  The Hamiltonian of the system, to be used in equation 2.2, is given by: H = gµB (S1 · B + S2 · B) + gµB S1 · ∑ A1 j I1 j + gµB S2 · ∑ A2 j I2 j j  (2.8)  j  Where the first term is the Zeeman Hamiltonian, which is the Hamiltonian for a particle with spin in an external magnetic field, and accounts for the geomagnetic field in this case. The second term is the so called hyperfine coupling [3], which is the energy split due to the coupling between the electron spin and each of the nuclear spins of the molecule. The physical origin of this term is that the nuclear spin, as any magnetic dipole moment, produces a magnetic field, which is felt by the electron spin, which is another dipole moment. Here g is the g-factor, which is approximately two for electron spin. µB is the Bohr magneton which is a natural unit for expressing the electron magnetic dipole moment. Its value is 9.27 × 10−24 JT −1 in SI units. B is the Earth magnetic field (or any magnetic field in a more general case). S1 and S2 are the electron spin operators defined in equa5  tion 2.5. Im j are the nuclear spin operators associated with electron m. (So each electron only interacts with the spins of its respective nuclei). They are defined in a similar way than the electron spin operators. Am j are the hyperfine coupling constants. The sum is over all the nuclear spins associated with each electron. Therefore, the first principles approach to model the radical pair mechanism of magnetoreception is to solve equation 2.2 with the Hamiltonian given by equation 2.8 then use equations 2.6 and 2.7 to extract the desired information. Some simplifications can be made. For example, if the singlet and triplet decay rates are the same (that is, kt = ks = k), and we assume that all the radical pairs are initially in a singlet state, so the density matrix at time zero is:  ρ (0) =  QS N  (2.9)  Where N is the number of spin states. The solution of equation 2.2 is then:  ρ (t) =  1 = exp[−iHt/¯h]Qs exp[iHt/¯h] N  (2.10)  Then, after inserting equation 2.10 into equation 2.6 we get an analytical expression for the triplet probability as a function of time: T (t) =  1 − 4N 4N T s e kt ∑ ∑ Qmn Qmn cos[(ωm − ωn )t] N m=1 n=1  (2.11)  Where ωi is the energy of the eigenstate i divided by h¯ . And, after using equation 2.7 to determine the triplet yield, we get the following expression: ΦT =  1 4N 4N T s k2 Q Q ∑ ∑ mn mn k2 + (wm − wn)2 N m=1 n=1  (2.12)  There is, unfortunately, a big practical difficulty in implementing this scheme: the size of the density matrix is equal to ∏ ni , where ni is the number of spin degrees of freedom of each particle in the system. A particle with spin n has 2n + 1 spin degrees of freedom. Hence just the two electrons, which have spin 1/2, will need a 4x4 density matrix. Adding a single 1/2 nuclear spin will increase the density matrix to 8x8. If we have 10 spin 1/2 nuclei, the density matrix would be of size 4096 x 4096. In our system, one of the two radicals may be superoxide (which does not 6  contain nuclear spin), and the other radical is FAD, which contains approximately 80 atoms. If all the FAD nuclei were spin 1/2 (Which they are not, some are spin zero, some are spin one, but this is just an estimate), then the size of the necessary density matrix would be of the order of 1024 × 1024 . The matrix would be much larger if the first radical were tryptophan rather than superoxide. And that still ignores the possible effect that all the other atoms in cryptochrome (around 4000) may have. There have been efforts to reduce the size of the matrix using group theory, by grouping atoms into independent terms of larger spin [29]. For example, 10 spin 1/2 nuclei, which by themselves (ignoring the electrons) would be represented by a 1024 x 1024 matrix are grouped into a statistical mixture of five particles, with spins five, four, three, two and one, respectively. The largest matrix corresponds to spin five, which would only be of size 11x11. So a 1024 matrix is converted into a weighted average of 11, 9, 7, 5 and 3 density matrices. Even with this simplification, the density matrices of realistic molecules are impossibly large to manage. Intuitively, the reason why the mechanism works can be viewed as follows: each of the electrons feels an effective magnetic field that is the sum of the Earth magnetic field and the field produced by the hyperfine coupling. This causes the electrons to precess at the Larmor frequency, which is proportional to the field magnitude. Now, the Earth field is the same for both electrons, but the hyperfine coupling field is different for each of them, so they will precess at different frequencies. This different precession frequency will continuously change their relative orientation, which in the quantum mechanical sense mean they change between singlet and triplet states. Changing the Earth magnetic field changes the field of each electron by the same magnitude, but, since their hyperfine fields are different, the ratio of their fields, and hence of their precession frequency, also changes, allowing a change in the Earth field to have an effect in the system. This interpretation form the basis of the so called semiclassical approximation [23]. In this approximation the nuclear spins are added as classical vectors, and the electron spins are allowed to precess around the resultant nuclear spin vector, while keeping the later fixed. That is, it treats the quantum mechanical electron precession phenomena as a classical vector precession. The static nature of the total nuclear spin is also an approximation, since actually both spins precess around a common 7  axis, but this axis approaches the total nuclear spin vector as the size of the radical increases. We will not discuss any further the semiclassical approximation in this work.  8  Chapter 3  Results and Discussion I: Coherent Evolution We are now going to discuss the coherent evolution of the two radical pair system. By coherent evolution we mean the interconversion between singlet and triplet state. This is effectively measured by the triplet probability (the probability that the system is in a triplet state as a function of time). As discussed in the previous chapter, if the decay rate is the same for singlet and triplet radical pairs, and assuming the the system is initially in a singlet state, the triplet probability is given by: T (t) =  1 − 4N 4N T s e kt ∑ ∑ Qmn Qmn cos[(ωm − ωn )t] N m=1 n=1  (3.1)  To study this better, it is convenient to set k = 0, that is, ignore decay into products and focus exclusively on singlet - triplet conversion. Therefore, all the graphics in this article must be multiplied by a negative exponential factor in order to represent the real system. Note that the time dependence comes exclusively from the cos[(ωm − ωn )t] term, where the ωi are the eigenenergies of the system (Divided by the Planck constant). We will now illustrate how to calculate the energy eigenvalues for a simple system in which one radical contains a single spin 1/2 nuclei and the other radical contain no nuclear spin. 9  The Hamiltonian is given by: H = gµB (S1 · B + S2 · B) + gµB AS2 · I  (3.2)  Where we assumed that the single nuclear spin is associated with electron 2. Since each electron interact only with its nuclei and the Hamiltonian does not contain an explicit electron-electron interaction term, we can calculate the energy of each electron-nuclei subsystem individually, and the total energy is simply the sum if the two. The Hamiltonian of electron one is a simple Zeeman Hamiltonian in an external field B, since it has no associated nuclei and therefore no hyperfine interaction. If we assume that the field points in the z direction, the Hamiltonian can be expressed as: 1 H1 = gµb B 2  1  0  0 −1  .  (3.3)  so the energy eigenvalues are + 12 gµb B and − 12 gµb B, where g is the g-factor  and µb is the Bohr magneton.  Electron two is more complex since it contains both a Zeeman term and a hyperfine term. Besides it is a two particle system (electron and nuclei) so its representation requires a 4 × 4 matrix. The term for electron two is more complex and contains both a Zeeman and a hyperfine interaction. Beside, it contain both a nuclear and an electron spin, so it requires a representation in terms of a 4 × 4 matrix. The z-component spin matrix of the electron is given by σz ⊗ I where σz is the standard 2× 2 Pauli matrix, and I is a 2× 2 identity matrix. The spin matrix of the z direction of the nuclear spin is given by I ⊗ σz . The definitions of the electron and nuclear spin can be inverted as long as we are consistent through the procedure. With this definitions the electron and nuclear z-spin matrices are given by:  10    1 0  0  0    1 0 0  1  0 −1 0 Iz =  2  0 0 1 0 0 0   0  . 0   −1  0  0  . 0   −1       0 1 0 1 Sz =   2  0 0 −1 0 0    0  (3.4)  (3.5)  Similarly, the x and y components of the electron and nuclear spin are: 0 0 1 0    0 0 0 1  1 .  Sx =  2 1 0 0 0   0 1 0 0   0 1 0 0   1 0 0 0  1 .  Ix =  2 0 0 0 1   0 0 1 0   0 0 −i 0   0 0 0 −i  1 .  Sy =  2 i 0 0 0   0 i 0 0   0 −i 0 0   i 0 0 0  1 .  Iy =  2  0 0 0 −i   0 0 i 0  (3.6)  (3.7)  (3.8)  (3.9)  And so the Zeeman plus hyperfine Hamiltonian of electron two is given by (We assumed that the hyperfine coupling constant is isotropic here):  11        gB + A  0  0  0  0  gB − A  2A  0  0  2A  −gB − A  0  0  0  0  −gB + A  1 0  0    0 1 0 1 H2 = µb gB   0 0 −1 2  0 0    1  H2 = µb  2    0  0  1  0  0  0     0   + 1 µb A  0 −1 2 0  0 2 −1 0 0   2  −1 0 0 0 1        .      .    (3.10)  (3.11)  Diagonalizing this matrix we get the following energy levels: 1 2 µb (A − gB) 1 2 µb (A + gB) 1 2 µb (−A − 1 2 µb (−A +  4A2 + g2 B2 ) 4A2 + g2 B2 )  So that the energy levels of the whole system are: 1 2 µb A. Double degenerate. 1 2 µb (A − 2gB) 1 2 µb (A + 2gB) 1 4A2 + g2 B2 ) 2 µb (−A + gB − 1 4A2 + g2 B2 ) 2 µb (−A − gB − 1 4A2 + g2 B2 ) 2 µb (−A + gB + 1 4A2 + g2 B2 ) 2 µb (−A − gB +  There are several things to note here. First, if the external field is zero, there are only two energy levels:  1 2 µb A  and − 32 µb A. Therefore, coherent evolution of  the system is a perfect sinusoidal function with a single frequency 2µb A. From this point of view, it is clear that what the external magnetic field does is to break  12  degeneracies in the system, hence making the coherence evolution a sum of various cosine terms with different frequencies. In a realistic manifestation of this system in the eye of a bird, the external field is much smaller than the hyperfine coupling, since the Earth magnetic field is much weaker as the magnetic field due to the total hyperfine coupling between the electron and the protein nuclei. The Earth magnetic field would then break the degeneracies present in the system, but the energy levels would be still grouped around 12 µb A and − 32 µb A. The coherent evolution would then be the sum of several cosine terms, with frequencies divided in two groups, and the frequencies within each group will be very close to each other, but not equal. The triplet probability vs time would then show a pulse behavior, as is the case when we add two sinusoidal term with close, but not equal, frequency. The opposite case, where the magnetic field is much larger than the hyperfine coupling, is investigated here for completeness, despite not being very relevant for the modeling of a realistic radical pair system. In this case the energy levels collapse in three groups: One around zero, and two around 12 µb gB and − 21 µb gB, respectively. In this case we also expect a pulse behavior in the triplet probability, because we are grouping cosine terms with very similar, but different frequencies. Figures 3.1 to 3.17 illustrate the coherent evolution of the radical pair for different system variables. In cases where the decay rates are different than zero, they are given in units of 106 s−1 . All the hyperfine couplings in those graphs are isotropic. The decay constant is taken to be zero, to illustrate more clearly the oscillation between singlet and triplet states (the so called coherent evolution). The numerical values were calculated using equation 2.11. Figures 3.1 to 3.4 show plots for a magnetic field of one Gauss and different hyperfine coupling constants. This first thing to notice is that the oscillation frequency increases as the hyperfine coupling constant increases. This is evident from the energy eigenstates derived previously, and the fact that the plot is a sum of cosine functions with frequency equal to the energy difference between eigenstates. The magnitude of the energy increases when either the field or the hyperfine coupling constant increases. Figure 3.4 shows the pulse behavior alluded earlier, which was expected when the hyperfine constant is much larger than the magnetic field. 13  Figures 3.5 and 3.6 show the case when the field is larger than the hyperfine constant. The pulses are also visible here, and look better defined than in the case where the hyperfine coupling is larger than the field. Note how the pulse period increases when the external field increases: Figure 3.5 with a magnetic field of ten Gauss present a pulse period of about 3.5 × 10−5 s, while figure 3.6, with a field of 100 Gauss shows a pulse period of 3.5 × 10−4 s. Therefore a 10x increase in the external field produced a 10x increase in the pulse period. Although not visible in the graph, the individual oscillation period decreased by a a factor of ten. Figure 3.7 show the case when both the field and the hyperfine coupling are increased. Comparing with figure 3.1 we see that the two graphs have the same shape, but the period of 3.1 is ten times greater, as can be seen from the time scale. Other simulations we did (not shown here) confirm that this is a general result: if we increase both the field and the hyperfine constant by a factor of n, the triplet probability plot has the same shape, but with the oscillation period decreased by the same factor of n. In figures 3.8 to 3.14, the system is changed to a slightly larger system: this time one radical has three spin 1/2 nuclei, and the other radical again has no nuclear spin. This is done in part to analyze which of the qualities of the single spin system are likely to be present in more realistic systems. We should note that when we changed the hyperfine coupling constants, we changed all three at the same time. It is obviously also possible to vary individually each hyperfine constant, but that possibility was not explored here. First, comparing figure 3.8 and figure 3.13, it can be seen that increasing both the magnetic field and the hyperfine coupling constant keeps the shape of the triplet probability graph, but increases the frequency. In this case, switching the field from one to ten gauss and the three hyperfine couplings also from one to ten gauss, increased the frequency by a factor of one hundred. It is reasonable that the increasing factor is larger than in the one spin nuclei system, since we now have a larger total hyperfine coupling, although at present we do not have an explanation of why this factor must be 100. And by analyzing the case when either the hyperfine constant or the Earth field is much larger than the other (figures 3.9, 3.11, 3.12 and 3.14, respectively, we see that we still observe the pulse behavior, although is much less clear than in the single spin case. It is likely that this behaviors would be mostly destroyed in a larger system. 14  Finally, in figures 3.15 to 3.17 we treat again the single spin one half system, but now we introduce decay rate constants. The magnetic field and the hyperfine coupling were fixed at one Gauss for these graphs. Since we tried using different singlet and triplet decay rate constants, equation 2.11 cannot be used, as it assumes that both decay rates are the same. Rather, we numerically integrated the Stochastic Liouville equation using a Runge Kutta algorithm. There is something that should be said about this graph. The concept of triplet probability and singlet probability strictly only make sense when there is no decay: in this case, at any specific time, the triplet probability is one minus the singlet probability, that is, the system is either a singlet or a triplet. But when there is decay, the sum of the singlet and triplet probabilities is no longer one, rather, it decreases with time. This quantities can be interpreted as the fraction of the radicals that are in a singlet or triplet states, taken with respect to the original number of radicals. As time progresses, the total number of radicals decrease, and so does the number of radicals in triplet or singlet states. As can be seen in the graphs, there is no significant difference in the triplet probability if we choose ks=1 and kt=0 or kt=0 and ks=1. The case where both kt=1 and ks=1 decays faster, but this is obviously because the total decay rate is larger: The system can decay from either a singlet or a triplet state.  15  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  0.2  0.4  0.6  0.8  time (s)  Figure 3.1: Single Spin one half, External field of one Gauss, hyperfine coupling one gauss  16  1 −5  x 10  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  0.2  0.4  0.6  0.8  time (s)  Figure 3.2: Single Spin one half, External field of one Gauss, hyperfine coupling ten gauss  17  1 −5  x 10  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  0.2  0.4  0.6  0.8  time (s)  Figure 3.3: Single Spin one half, External field of one Gauss, hyperfine coupling 100 gauss  18  1 −5  x 10  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  1  2  3  4  time (s)  Figure 3.4: Single Spin one half, External field of one Gauss, hyperfine coupling 100 gauss  19  5 −5  x 10  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  1  2  3  4  time (s)  Figure 3.5: Single Spin one half, External field of ten Gauss, hyperfine coupling one gauss  20  5 −5  x 10  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  1  2  3 time (s)  4  5  Figure 3.6: Single Spin one half, External field of 100 Gauss, hyperfine coupling one gauss  21  6 −4  x 10  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  0.2  0.4  0.6  0.8  time (s)  Figure 3.7: Single Spin one half, External field of ten Gauss, hyperfine coupling ten gauss  22  1 −6  x 10  0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  0.2  0.4  0.6  0.8  time (s)  Figure 3.8: Three Spin one half, External field of one Gauss, hyperfine coupling one gauss  23  1 −5  x 10  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  0.2  0.4  0.6  0.8  time (s)  Figure 3.9: Three Spin one half, External field of one Gauss, hyperfine coupling ten gauss  24  1 −5  x 10  0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  0.2  0.4  0.6  0.8  time (s)  Figure 3.10: Three Spin one half, External field of one Gauss, hyperfine coupling 100 gauss  25  1 −5  x 10  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  0.2  0.4  0.6  0.8  time (s)  Figure 3.11: Three Spin one half, External field of one Gauss, hyperfine coupling 100 gauss  26  1 −4  x 10  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  0.5  1 time (s)  1.5  Figure 3.12: Triplet Probability vs. time, Three Spin one half, External field of ten Gauss, hyperfine coupling one gauss  27  2 −5  x 10  0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  0.5  1 time (s)  1.5  Figure 3.13: Three Spin one half, External field of ten Gauss, hyperfine coupling ten gauss  28  2 −7  x 10  1 0.9 0.8  Triplet probability  0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  0  1 time (s)  Figure 3.14: Three Spin one half, External field of 100 Gauss, hyperfine coupling one gauss  29  2 −4  x 10  0.7  0.6  Triplet probability  0.5  0.4  0.3  0.2  0.1  0  0  0.2  0.4  0.6 time (s)  0.8  1  Figure 3.15: Triplet Probability vs. time with decay constants ks=0 and kt=1  30  1.2 −5  x 10  0.7  0.6  Triplet probability  0.5  0.4  0.3  0.2  0.1  0  0  0.2  0.4  0.6 time (s)  0.8  1  Figure 3.16: Triplet Probability vs. time with decay constants ks=1 and kt = 0  31  1.2 −5  x 10  0.5 0.45 0.4  Triplet probability  0.35 0.3 0.25 0.2 0.15 0.1 0.05 0  0  0.2  0.4  0.6 time (s)  0.8  1  Figure 3.17: Triplet Probability vs. time with decay constants ks=1 and kt=1  32  1.2 −5  x 10  Chapter 4  Results and Discussion 2: Triplet Yield The triplet yield is calculated with equation: ΦT =  k2 1 4N 4N T s Qmn Qmn 2 ∑ ∑ N m=1 n=1 k + (wm − wn )2  (4.1)  In this case, it is the sum of a term with the form k2 /(k2 + (wm − wn )2 ) for each combination of eigenenergies wm and wn . The energy expressions for the single spin 1/2 system were found in the previous chapter: 1 2 µb A. Double degenerate. 1 2 µb (A − 2gB) 1 2 µb (A + 2gB) 1 4A2 + g2 B2 ) 2 µb (−A + gB − 1 4A2 + g2 B2 ) 2 µb (−A − gB − 1 4A2 + g2 B2 ) 2 µb (−A + gB + 1 4A2 + g2 B2 ) 2 µb (−A − gB +  In our first set of plots we are going to show the triplet yield as a function of the magnitude of the external field. We remember the interpretation we give 33  in the previous chapter, when we saw that the field breaks degeneracies in the eigenenergies of the radical pair system. This activates more sinusoidal terms in equation 2.11, which means that singlet triplet mixing is enhanced. Since we are assuming that our system is initially a singlet, we can expect that increasing the magnitude of the magnetic field will increase the triplet yield. From another point of view, turning on a field will add more terms to the sum in equation 4.1. Note however that the energies wi appear in the denominator, so as the field increases, most of the terms in the sum will tend to zero. There is an exception to this, though: The double degenerate energy 21 µb A, which do not depend on the external field. This will give wm − wn = 0 which contributes a constant term to the sum 4.1, that is independent of the magnitude of the field. Hence, we expect that the triplet yield will increase for small magnetic fields, but will eventually stabilize to a constant value as the field increases. This is shown in figures 4.1 and 4.2. Figure 4.1 shows the case when the hyperfine coupling constant (HCC) is one Gauss. It clearly show that there is a field intensity for which the triplet yield is a maximum, and beginning from that point it decreases until it reaches its stable value. This behavior is visible for all plotted values of the decay rate. It is also shown that the overall triplet yield decreases as the rate increases. This is because the system is initially a singlet, therefore, for early times, all products decay from singlet state and do not contribute to the triplet yield. A higher decay constant means that by the time the system is in a triplet state for the first time, a significant portion of the radicals have already decayed. Figure 4.2 show the case when the HCC is 100 Gauss. This plots show that the the maximum disappear for large values of the HCC, instead, it simply increases for small fields until it reaches an stable value. It is also clear that the effect of the decay rate is reduced. This last observation can be explained with the behavior we saw in the last section: the frequency of coherent evolution increases as either the field or the HCC increases. This means that that, if the system is originally a singlet, it becomes a triplet faster, and so the ”singlet loss” effect is greatly reduced. Figures 4.3 and 4.4 show the triplet yield for a three spin 1/2 nuclei system. The HCC is the same for the three nuclei. Figure 4.3 show that for a small hyperfine coupling the result is quite different than the single nuclei case. Here we find that we have both a maximum and a minimum, a feature that disappears as 34  we we increase the decay rate. On the other hand, figure 4.4 shows that for a hyperfine coupling of 100 Gauss, the results are quite similar to the single nuclei system. The overall yield is higher in the three spin case, but the difference is less than 3 percent. What we found is that for a fixed number of nuclei, the triplet yield converged to a fixed functional form as the HCC increases. For example, the triplet yield looks practically the same for HCC of 100 and 1000 Gauss, with the three nuclei system having a slightly larger yield than the one nuclei system. It would be interesting to investigate if increasing the number of nuclei (and thus the total hyperfine coupling constant) would give a similar convergence. If this is true, it will not be necessary to spend computing resources modeling more accurately the real system, since a single spin 1/2 system with a high hyperfine coupling constant would give the same result. Note that this is only true for the triplet yield as a function of the magnetic field, not for the coherent evolution, because, as we saw in the previous section, the coherent evolution oscillation frequency increases as the HCC increases. It should also be noted that this possible approximation is not the same as the one presented in [14] or [24], which were approximations for the evolution of the triplet probability. So far, we have investigated the effect of the magnitude of the Earth magnetic field in the triplet yield. However, since the system is hypotezised to be used by birds to get orientational information, there should be an effect on the triplet yield caused by changing the direction of the magnetic field. For this, it is necessary that the hyperfine coupling constants are anisotropic. To see this, consider the derivation of the energy eigenstates given in the previous chapter. We chose the magnetic field to point in the z direction, but that choice is arbitrary. If the hyperfine coupling constant is isotropic, there is no preferred direction, so we can arbitrary choose the z axis to be the direction of the external magnetic field. So, if we rotate the magnetic field we can rotate the z axis with it, and so for each direction of the field the energy eigenvalues are going to be the same. A glance at equation 4.1 shows that the triplet yield can only change if the eigenvalues wi change, so rotating the magnetic field would have no effect on the triplet yield. On the other hand, if the hyperfine coupling is anisotropic, then once we choose a direction for the z-axis, that direction must remain fixed as we rotate the field. The derivation of analytic expression for the energy eigenvalues poses no concep35  tual problem, however, the calculations become too cumbersome, so we will limit ourselves with treating the problem numerically. The results are shown in figures 4.5 to 4.7. We use a HCC of the form Ax = n, Ay = 0 and Az = n where n is one, thirty or sixty Gauss. We can again see that for small HCC there is a large influence from the value of the decay rate, but this disappears as the HCC is increased. Also, comparing the results HCC of 30 and 60 gauss, the effect of the magnitude of the HCC itself begins to fade. Finally, figures 4.8 to 4.10 show the effect of varying the singlet and triplet decay rates in the directional triplet yield. It is not possible to use equation 4.1 in these calculations, since it is an approximation only applicable for constant decay rates. Instead we integrated the stochastic Liouville equation using a Runge Kutta method, and then numerically integrated the result over time to obtain the triplet yield for each angular value of the magnetic field. Due to technical limitations the oscillatory frequency of the coherent evolution cannot be too high in order to do the numerical integration, so it was not possible to only use a large hyperfine constant. All the results were performed with an anisotropic HCC of one Gauss. The decay rates are as always given in units of 106 s−1 . In order to compare with figure 4.5, in which both the singlet (ks) and the triplet (kt) decay rates are equal to one, we choose combinations of singlet and triplet decay rates so that ks+kt=2. Note that we did not include kt=0, since in this case the triplet yield is exactly zero for all angles. Even though the shape of the curves are similar, there are two important differences: First, the higher kt the higher is the triplet yield. For example, for kt=2, ks=0 the triplet yield in in the range 0.998 to 0.999, while for kt=0.5, ks=1.5 we get the range 0.43 to 0.49. The second difference is that the triplet yield spans a larger range the smaller the triplet decay rate is. Since it is changes in the triplet yield what birds would hypothetically use for navigation, the second difference is the most important. It means that the mechanism is more sensible if the singlet decay constant is much smaller than the triplet decay constant. However, since it is computationally advantageous to set ks=kt, since using equation 4.1 is much cheaper than integrating the equation of motion numerically, it would be useful to find a relationship between the triplet yield range and the difference in ks and kt, thus making it possible to solve for the ks=kt case and apply the correction a posteriori. We did not pursued this line in the present work. 36  0.55  Triplet yield  0.5  0.45  k=1 k = 1.67 k = 3.33  0.4  0.35  0  2  4 6 magnetic field (Gauss)  8  Figure 4.1: Single Spin one half, hyperfine coupling one gauss  37  10  0.7  0.65  Triplet yield  0.6 k=1 k = 1.67 k = 3.00  0.55  0.5  0.45  0.4  0.35  0  5  10 magnetic field (Gauss)  15  Figure 4.2: Single Spin one half, hyperfine coupling 100 gauss Now we have the three spin half systems.  38  20  0.6 k=1 k = 1.67 k = 3.33  0.55  Triplet yield  0.5  0.45  0.4  0.35  0  2  4 6 magnetic field (Gauss)  8  Figure 4.3: Three Spin one half, hyperfine coupling one gauss  39  10  0.65  Triplet yield  0.6  0.55  0.5  0.45  0.4  0  5  10 magnetic field (Gauss)  15  Figure 4.4: Three Spin one half, hyperfine coupling ten gauss  40  20  0.5  Triplet yield  0.45  0.4  0.35  k=1 k = 1.67 k=3  0.3  0.25  0  0.5  1 1.5 2 2.5 azimuthal angle with respect to x axis  3  Figure 4.5: Angle Dependence, Field one Gauss, Hyperfine Coupling one Gauss  41  3.5  0.75  0.74  Triplet yield  0.73  0.72  0.71  0.7 k=1 k=1.67 k=3  0.69  0.68  0  0.5  1 1.5 2 2.5 azimuthal angle with respect to x axis  3  Figure 4.6: Angle Dependence, Field one Gauss, Hyperfine Coupling thirty Gauss  42  3.5  0.75  0.74  0.73  0.72  0.71  0.7  k=1 k = 1.67 k=3  0.69  0.68  0  0.5  1  1.5  2  2.5  3  Figure 4.7: Angle Dependennce, Field one Gauss, Hyperfine Coupling sixty Gauss  43  3.5  0.9992  0.999  Triplet yield  0.9988  0.9986  0.9984  0.9982  0.998  0  0.5  1 1.5 2 2.5 azimuthal angle with respect to x axis  Figure 4.8: Angle Dependence, Rate Constants kt = 2, ks = 0  44  3  3.5  0.905 0.9 0.895  Triplet yield  0.89 0.885 0.88 0.875 0.87 0.865 0.86  0  0.5  1 1.5 2 2.5 azimuthal angle with respect to x axis  Figure 4.9: Angle Dependence, Rate Constants kt = 1.5, ks = 0.5  45  3  3.5  0.49  0.48  Triplet yield  0.47  0.46  0.45  0.44  0.43  0  0.5  1 1.5 2 2.5 azimuthal angle with respect to x axis  Figure 4.10: Angle Dependence, Rate Constants, kt = 0.5, ks = 1.5  46  3  3.5  Chapter 5  Conclusion In this work we have examined a radical pair model for magnetoreception. As we saw, it is in possible for the direction of the magnetic field to have an influence on the triplet yield of a radical pair reaction, which can in principle be used as a magnetoreception mechanism. Furthermore, even if our simulations were performed on very simplified systems, it is unlikely that the results would be significantly different for the real system. This is because the triplet yield tends tends to a constant functional form as we increase the size of the hyperfine coupling constant. In our graphs the maximum variation of the orientational triplet yield in our simulations was less than one percent. While it is true that we did not exhaustively investigated all possible combinations of decay rate constants and hyperfine coupling constants, it is still plausible that the directional effect on the triplet yield will be small. it is worthwhile then to ask if such a small effect can be used as a sensorial mechanism. Since it is speculated that the magnetoreception mechanism is linked with the visual sense [10], meaning that the bird would eventually see bright and dark spots depending on the position of the field, it is worthwhile then to compare the triplet yield difference with the sensitivity of vision. In a landmark experiment, Hecht, Shlaer and Pirenne determined that as little as five to eight photons are enough to trigger a visual response in humans [9]. The vision of most birds is better than the human vision, since they contain a higher density of photoreceptor cells [28]. An average human has about 125 millions of photoreceptor cells in their retina. Birds have a higher density of photoreceptor cells [28], but less 47  surface area on their retinas, so lets say for the argument that they have 100 million photoreceptor cells. If there is one cryptochrome molecule in each cell, we have 100 millions radical pairs. One percent of this is 1 million. Since eight photons, where each photon strikes a different cell are enough to trigger a visual response, it is reasonable to assume than the million photoreceptor cells that change during the magnetoreception mechanism could in principle give a sensorial response. Furthermore, if this mechanism is escencial for survival, natural selection would have adjusted the parameters to optimize its function. We believe that in order to push forward the research in this topic, new extensive experimental studies are needed. While there is an extensive bibliography on behavioral studies in birds that can give hints on the structure of the mechanism cite2 [21] [32], there are still comparatively few experimental research at the molecular levels. It is known that cryptochrome is present in the eyes of migratory birds [18]. However, magnetite particles (the competing magnetoreceptor theory) have also being found in birds retina [5], which may suggest a combined radical pair - magnetite mechanism. Evidence of FAD - Tryptophan radical pairs whose formation is triggered by the absorption of blue light has been found [16]. There is also recently one recent paper that claims to have found FAD - superoxide radical pairs [17]. However, none of these works show conclusively that the radical pairs are related to magnetoreception. Some work can stil be done on a theoretical level. First, it will be nearly impossible to solve a realistic system using the stochastic Liouville equation, but, as we saw in our results, it may not be necessary. Further simulations are needed to determine precisely the asymptotic behavior of the triplet yield function as the number of nuclei are increased. Our results suggest that there is indeed an asymptotic behavior. It is also important to determine if superoxide binds to cryptochrome with enough frequency to support the proposed mechanism. An important analogous is the interaction of superoxide with the protein superoxide dismutase, which has been extensively studied. It is known that in that case, the reaction is diffusion limited, and is encaced by an electrostatic funnel [7]. It is therefore of interest to investigate whether such an analogous mechanism also exist in cryptochrome. On a deeper theoretical level are the purely quantum mechanical effects such 48  as decoherence or the quantum Zeno effect that may have influence on this system [1] [13]. Insights gained in this area may have applications in the growing field of quantum computation, in the sense that the radical pair mechanism would be a room temperature quantum computer [11]. Furthermore, this is among the first proposed mechanisms in which a non trivial quantum phenomena would play a fundamental role in a biological function. Quantum phenomena in biology is a relatively new area of research, and we hope that further study of this mechanism will provide valuable insights for this emergent field.  49  Bibliography [1] J. Cai, G. G. Guerreschi, , and H. J. Briegel. Quantum control and entanglement in a chemical compass. Physical Review Letters, 2010. → pages 49 [2] A. R. Cashmore, J. A. Jarillo, Y.-J. Wu, and D. Liu. Cryptochromes: Blue light receptors for plants and animals. Science, 1999. → pages 2 [3] C. Cohen-Tannoudji, B. Diu, and F. Laloe. Quantum Mechanics. Wiley, 2006. ISBN 0471569526. → pages 5 [4] J. S. Donald McQuarrie. Physical Chemistry: A Molecular Approach. University Science Books, 1 edition, 1997. ISBN 0935702997. → pages 4 [5] G. Fleissner, E. Holtkamp-Rtzler, M. Hanzlik, M. Winklhofer, G. Fleissner, N. Petersen, and W. Wiltschko. Ultrastructural analysis of a putative magnetoreceptor in the beak of homing pigeons. The Journal of Comparative Neurology, 2003. → pages 48 [6] R. J. Gegear1, L. E. Foley1, and A. C. andd Steven M. Reppert. Animal cryptochromes mediate magnetoreception by an unconventional photochemical mechanism. Nature, 2010. → pages 1 [7] E. D. Getzoff, D. E. Cabelli, C. L. Fisher, H. E. Parge, M. S. Viezzoli, L. Banci, and R. A. Hallewell. Faster superoxide dismutase mutants designed by enhancing electrostatic guidance. Nature, 1992. → pages 48 [8] B. Giovani, M. Byrdin, M. Ahmad, and K. Brettel. Light induced electron transfer in a cryptochrome blue light photoreceptor. Nature Structural Biology, 2003. → pages 2 [9] S. Hecht, S. Shlaer, , and M. H. Pirenne. Energy, quanta and vision. The Journal of General Physiology, 1942. → pages 47  50  [10] D. Heyers, M. Manns, H. Luksch, O. Gntrkn, and H. Mouritsen1. A visual pathway links brain structures active during magnetic compass orientation in migratory birds. PLoS One, 2007. → pages 47 [11] J. Jonesa and P. H. b. Spin-selective reactions of radicalpairs act as quantum measurements. Chemical Physica Letters, 2010. → pages 49 [12] J. Kirschvinka, M. Walkerb, and C. Diebel. Magnetite-based magnetoreception. Current Opinion in Neurobiology, 2001. → pages 1, 2 [13] I. K. Kominis. Quantum zeno effect explains magnetic-sensitive radical-ion-pair reactions. Physical Review E, 2009. → pages 49 [14] K. S. Krane. Introductory Nuclear Physics. Wiley, 3 edition, 1987. ISBN 047180553. → pages 2, 35 [15] R. Kubo. Stochastic liouville equations. Journal of Mathematical Physics, 1962. → pages 5 [16] M. Liedvogel, K. Maeda, K. Henbest, E. Schleicher, T. Simon, C. R. Timmel, P. J. Hore, and H. Mouritsen. Chemical magnetoreception: Bird cryptochrome 1a is excited by blue light and forms long-lived radical-pairs. PLoS One, 2007. → pages 48 [17] P. Mller and M. Ahmad. Light-activated cryptochrome reacts with molecular oxygen to form a flavinsuperoxide radical pair consistent with magnetoreception. The Journal of Biological Chemistry, 2011. → pages 48 [18] H. Mouritsen, U. Janssen-Bienhold, M. Liedvogel, G. Feenders, J. Stalleicken, P. Dirks, and R. Weiler. Cryptochromes and neuronal-activity markers colocalize in the retina of migratory birds during magnetic orientation. Proceedings of the National Academy of Science, 2004. → pages 48 [19] H. Mouritsen1 and T. Ritz. Magnetoreception and its use in bird navigation. Current Opinion in Neurobiology, 2005. → pages 2 [20] T. Ritz, S. Adem, and K. Schulten. A model for photoreceptor-based magnetoreception in birds. Biophysical Journal, 2000. → pages 1, 2, 3 [21] T. Ritz, P. Thalau, J. B. Phillips, R. Wiltschko, and W. Wiltschko. Resonance effects indicate a radical-pair mechanism for avian magnetic compass. Nature, 2004. → pages 1, 48 51  [22] J. J. Sakurai and J. Napolitano. Modern Quantum Mechanics. Addison Wesley, 2 edition, 2010. ISBN 0805382917. → pages 3 [23] K. Schulten and P. Wolynes. Semiclassical description of electron spin motion in radicals including the effect of electron hopping. Journal of Chemical Physics, 1977. → pages 7 [24] K. Schulten, H. Staerk, A. Weller, H.-J. Werner, and B. Nickel. Magnetic field dependence of the geminate recombination of radical ion pairs in polar solvents. Zeitschrift fr Physikalische Chemie, 1976. → pages 3, 35 [25] I. A. Solov’yov and K. Schulten. Magnetoreception through cryptochrome may involve superoxide. Biophysical Journal, 2009. → pages 2 [26] I. A. Solovyov, D. E. Chandler, and K. Schulten. Magnetic field effects in arabidopsis thaliana cryptochrome-1. Biophysical Journal, 2007. → pages 2, 4 [27] C. Timmel, U. Till, B. Brocklehurst, K. Mclauchlan, and P. Hore. Effects of weak magnetic fields on free radical recombination reactions. Molecular Physics, 1998. → pages 3, 4 [28] J. A. Waldvogel. A bird’s eye view. American Scientist, 1990. → pages 47 [29] H. J. Werner, K. Schulten, and Z. Schulten. Theory of magnetic field modulated geminate recombination of radical ion pairs in polar solvents: Aplication to the pyrene-n, n-dimethylaniline system. The Journal of Chemical Physics, 1977. → pages 7 [30] R. Wiltschko and W. Wiltschko. Magnetic Orientation in Animals (Zoophysiology). Springer, 1 edition, 1995. ISBN 3540592571. → pages 1 [31] W. Wiltschko and R. Wiltschko. Magnetic compass of european robins. Science, 1972. → pages 1 [32] W. Wiltschko and R. Wiltschko. Magnetic orientation in birds. The Journal of Experimental Biology, 1996. → pages 1, 48  52  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0072778/manifest

Comment

Related Items