Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Gadhiwala, Nitya
Master of Science - MSc
University of British Columbia
Recent advances in single cell sequencing has led to the availability of gene expression data. We model the trajectories of cells in gene expression space using a diffusion and gradient drift process that also undergoes branching. We recover the law on paths of this process given time marginals using an entropy minimization problem. The optimization is over the set of all processes with the same branching mechanism and given marginals. We prove that among these the one which minimizes the relative entropy with respect to a reference branching Brownian motion is the ground truth. The above result has been proved earlier for models that don't consider branching. This thesis extends it for the case with branching.
Branching Brownian motion models forcell developmental trajectoriesbyNitya GadhiwalaBachelor of Mathematics (Honours), Indian Statistical Institute, 2021A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2023© Nitya Gadhiwala 2023The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Branching Brownian motion models for cell developmental trajectoriessubmitted by Nitya Gadhiwala in partial fulfillment of the requirements forthe degree of Master of Sciencein Mathematics.Examining Committee:Omer Angel, Professor, Department of Mathematics, UBCCo-supervisorGeoffrey Schiebinger, Assistant Professor, Department of Mathematics, UBCCo-supervisoriiAbstractRecent advances in single cell sequencing has led to the availability of geneexpression data. We model the trajectories of cells in gene expression spaceusing a diffusion and gradient drift process that also undergoes branching.We recover the law on paths of this process given time marginals usingan entropy minimization problem. The optimization is over the set of allprocesses with the same branching mechanism and given marginals. Weprove that among these the one which minimizes the relative entropy withrespect to a reference branching Brownian motion is the ground truth. Theabove result has been proved earlier for models that don't consider branching.This thesis extends it for the case with branching.iiiLay SummaryA cell, over the course of it's lifetime, traces out a path in gene expressionspace. Upon it's death, it divides into a random number of cells. Usingmodern technologies such as single cell sequencing, we can obtain snapshotsof these paths in gene expression space. We use optimal transport to fit amodel of branching diffusion with gradient drift to the above data. We usethe relative entropy with a reference branching diffusion as the cost function.We prove that the solution to the optimal transport problem recovers theground truth if all the marginals are available.ivPrefaceThe thesis consists of three chapters. The first chapter is the introductionwhere we give a broad overview of the biological motivation and the mathe-matical models for inferring cell development trajectories on gene expressionspace. The second chapter contains existing results on the variational charac-terization of the law of the diffusion given marginal constraints [13, Theorem4.1] and the reduction of this optimization problem from path space to a fi-nite dimensional product distribution on gene expression space [4, Theorem3.1]. The third chapter contains a new result where we extend [13, Theorem4.1] to the case with branching. For this we rely on tools such as Radon-Nikodym derivative from [2, Theorem 4.23] and stochastic integration toolsfor Branching Brownian motion [2, Theorem 4.19]. Our main result, Theo-rem 3.2.1 and proof are presented in chapter 3.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The model and literature . . . . . . . . . . . . . . . . . . . . 41.2 Overview of the results and layout . . . . . . . . . . . . . . . 82 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . 102.1 Optimal Transport . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Relative entropy . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 A variational characterization of the law of a diffusion . . . . 152.4 Trajectory inference using diffusion models . . . . . . . . . . 193 A variational characterization of the law of a branching dif-fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Limitations and Future Work . . . . . . . . . . . . . . . . . . 36Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37viList of Figures1.1 The above represents a two dimensional projection of G di-mensional gene expression space. Each of the axes representsthe number of molecules of the gene g1 and g2 (say). The pathdescribes the values along this two dimensional projection ofG over time. (a) represents the ground truth and t1 and t2are the times of branching. (b) represents the times T1 and T2when sequencing was done, and the black points denote thegene expressions of the cells observed. (c) and (d) representsthe tree structure that we estimate from the observations attimes T1 and T2 respectively. Note that the times t1 and t2correspond to the same times of branching as in the groundtruth (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 This figure is taken from [24]. The movement of the ball rep-resents the cell moving in gene expression space as it develops.The ball as it is pictured is an undifferentiated cell and as itmoves to the stable minimas, it specializes into different celltypes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 This figure is a representation of the optimal transport mapsbetween the different time points. Here Pt1,t2 is the optimaltransport coupling between Pˆt1 and Pˆt2 which are the empir-ical observations at time t1 and t2 respectively. The x axisrepresents time, the y axis represents the state of the cell ingene expression space which we assume to be Rd. . . . . . . 71.4 This figure contains the added lineage tree information. Usingthis we may estimate the branching mechanism. This estimatecan then be used in the branching diffusion model consideredin the relative entropy minimization problem. The x axisrepresents time, the y axis represents the state of the cell ingene expression space which we assume to be Rd. . . . . . . 7viiList of SymbolsN Set of natural numbersRd d dimensional Euclidean spaceR+ [0,∞)Td d dimensional torus∂s Derivative with respect to s∇ Gradient∆ Laplace operatorµ≪ ν µ is absolutely continuous with respect to νP(X) Set of probability measures on XM(X) Set of positive measures on XT♯µ = ν T is a pushforward of µ to νΠ(α, β) Set of couplings between measures α and βH(P ) Entropy of PH(P | Q) Relative entropy of P with respect to QC2(X,Y ) Set of continuously twice differentiable functions from X to YC∞(X,Y ) Set of smooth functions from X to YC(X,Y ) Set of continuous functions from X to YC2(X) Set of real valued continuously twice differentiable functions on XC(X) Set of real valued continuous functions on X{Bσt }t∈[0,1] Brownian motion with diffusivity σBσ Law of Brownian motion with diffusivity σBσ,x Brownian motion with diffusivity σ starting at x(Ft)t∈[0,1] Canonical filtrationMt Empirical measure describing cells alive at time tRti Marginal of R at time tiλ(t, x) Branching rate at time t and location xq Branching mechanismΦq(t,x) Generating function of q at (t, x)N (t) Set of cells alive at time tBBM(σ, q,R0) Branching Brownian motion with diffusivity σ,branching mechanism q and initial distribution R0viiiAcknowledgementsI would like to express my sincere gratitude to my supervisors, GeoffreySchiebinger and Omer Angel, for introducing me to this research topic andfor their consistent and attentive guidance throughout this project. Thanksto Parijat Chakraborty, Aden Forrow and Elias Ventre for useful discussionsduring the weekly group meetings.Last but not least, I would like to thank everyone who kept me sane in thelast two years.ixDedicationI dedicate this to my sister, Roma. You are right, math is hard.xChapter 1IntroductionEach cell in an organism has the same DNA but expresses different genes asit performs different functions. The RNA molecules in a cell provide us of asignature of what the cell does or what it is trying to do. Through a processcalled transcryption, the cell makes these RNA molecules of different genesfrom the DNA. These RNA molecules are later converted into proteins by aprocess called translation with the help of the ribosomes. One can observethe RNA molecules using modern day cell sequencing technology. Single cellRNA sequencing (scRNA-seq) is a process which takes a mass of cells/tissueas input and gives us counts of the different RNA molecules present in eachcell as output. It isolates individual cells, captures their RNA transcriptsand generates sequencing libraries. It is known that sc-RNA sequencing hasa lot of noise (See [11]) thus making the analysis harder.Cell development trajectories are analyzed to understand important ques-tions in biology like embryonic development and how cells destabilize withdisease or old age. Developmental biologists have used methods to tag cellsand trace lineages before but now it is possible to recover more complex re-lationships, like the full lineage tree of a population of cells. As mentionedabove, recently there have been new single cell measurement technologies(like single cell RNA sequencing ([12], [15]) and single cell ATAC-sequencing([3]) which provide a description of the gene expression profiles of cells. Fur-thermore, using CRISPR based lineage tracing ([16], [18], [19])) we can re-construct a cell lineage tree describing the shared ancestry of the cells. Moreprecisely, CRISPR-Cas 9 continuously mutates an array of synthetic DNAbarcodes, which are then inherited by the next generation of cells. By ana-lyzing the pattern of mutations in the barcodes, one can reconstruct a lineagetree.We discuss via a toy example how to infer the cell developmental trajectoriesfrom sc-RNA sequencing snapshots and CRISPR based lineage estimation.Consider the example in Figure 1.1. Suppose Figure 1.1(a) denotes theground truth i.e., here we have two branching events, one at time t1 and1Chapter 1. IntroductionFigure 1.1: The above represents a two dimensional projection of G dimen-sional gene expression space. Each of the axes represents the number ofmolecules of the gene g1 and g2 (say). The path describes the values alongthis two dimensional projection of G over time. (a) represents the groundtruth and t1 and t2 are the times of branching. (b) represents the timesT1 and T2 when sequencing was done, and the black points denote the geneexpressions of the cells observed. (c) and (d) represents the tree structurethat we estimate from the observations at times T1 and T2 respectively. Notethat the times t1 and t2 correspond to the same times of branching as in theground truth (a).2Chapter 1. Introductionanother at time t2. The figure shows the two dimensional projection of themovement of the cells in gene expression space. Using current technology, wecan only make measurements by lysing the cell. Note that the cells observedat T1 and T2 are different, however, they are assumed to follow the sametrajectory. Thus, the ground truth cannot be measured in it's entirety.In Figure 1.1(b) we make two cell measurements at time T1 and three attime T2 using single cell RNA sequencing. The gene expression profiles of acell is a vector whose entries denote the counts of the different types of RNAmolecules of each gene present inside the cell. Suppose we are observing Gtotal genes for each cell at each time point. The gene expression profiles ofall the cells observed at time T1 and T2 are given by PT1 and PT2 in (1.1).PT1 =Cell A Cell Bg1 7 8g2 3 6.........gG 2 10, PT2 =Cell i Cell ii Cell iiig1 6 14 0g2 9 7 3............gG 4 16 17.(1.1)Each column corresponds to a cell and each row corresponds to a gene (here,the genes are labelled g1, g2, . . . , gG). The entry for Cell i and gj correspondsto the count of Gene gj , found in Cell i in this scRNA-seq process. Forinstance, the above data tells us that we measured 6 RNA molecules of geneg2 in Cell B at time T1 and 17 RNA molecules of gene gG in Cell iii at timeT2. As these cells develop and specialize, we assume that they move in theG dimensional space.In addition to this, for any pair of cells observed at the same time, we cancompute the time of their most recent common ancestor by using CRISPR-Cas 9 based barcode editing technologies. These technologies continuouslymutate an array of synthetic DNA barcodes which are incorporated into thechromosomes and inherited across generations. By analyzing the pattern ofthese mutations in the barcodes, one can compute the time to the most recentcommon ancestor for a pair of cells and consequently reconstruct the entirelineage tree. In Figure 1.1(c), t0 is the starting time of cell development, T1is the time we observe the pair of cells and t1 is the computed time of themost recent common ancestor of the two cells. Figure 1.1(d) represents asimilar construction of the lineage tree for cells observed at a later time T2.The lineages are represented by the purple graph. It contains information31.1. The model and literatureregarding the time of latest common ancestor for any pair of cells.In summary, in Figure 1.1(b), we have the cell measurements at time T1 andT2 and in Figure 1.1(c) and (d), we have an estimate of the lineage tree.The inference problem is to combine these two sets of information to recon-struct the ground truth. The goal of the inference problem is to estimateor reconstruct the developmental trajectory of cells along with their ancestry.The availability of the above data has led to a flurry of work in inferring en-tire developmental trajectories ([6], [21], [25], [26], [20]). However, the mainchallenge with cell measurement technologies is that whenever we make anobservation, we need to lyse the cells, which prevents us from observing thesame cell at a later time. And so, it is now important to develop techniquesto infer developmental trajectories from these independent static snapshots.In the next section, we present a model inspired by the Waddington land-scape and describe the mathematical methods used to fit the dataset. Wealso do a brief survey of the literature.1.1 The model and literatureIn [24], Waddington suggests a simple model describing the movement ofdeveloping cells in gene expression space as balls rolling down a potentiallandscape, moving from a 'high potential' to a 'low potential'. Consider asurface like the one in Figure 1.2, where a ball placed on the surface wouldrun down, with random noise, towards a final end state on the bottom edge.We can think of each of the wells on the bottom edge as different types (i.e.,clusters in gene expression space) of specialized cells, like red blood cells orskin cells or cells in the spinal cord. We can think of the ball at the start ofit's motion to be a stem cell. We interpret it's motion along the landscape asdevelopment of the stem cell into one of these specializations. In early stagesof development, it may move around slowly between the different types ofcells but towards the end of the specialization process, it becomes more dif-ficult for cells to jump between different cell types as the potential barrierbecomes larger.Given the above, we will model the evolution of each cell over an infinitesimaltime interval dt by the stochastic differential equation in the gene expressionspace X (which is typically assumed to be a d dimensional Reimann manifold41.1. The model and literatureFigure 1.2: This figure is taken from [24]. The movement of the ball repre-sents the cell moving in gene expression space as it develops. The ball as itis pictured is an undifferentiated cell and as it moves to the stable minimas,it specializes into different cell types.without boundary)dXt = −∇Ψ(t,Xt)dt+ σdBt, (1.2)where Xt ∈ X gives the state of the cell at time t, Bt is the standard d-dimensional Brownian motion, Ψ : R+ × X → R is a potential that definesthe landscape and σ > 0 is refered to as the diffusion coefficient. As t varies,Xt describes the trajectory of the cells in X and (1.2) introduces a probabil-ity law on such trajectories.The gradient drift SDE model given by (1.2) cannot be used to model non-conservative oscillatory dynamics like cell cycles and circadian rhythms sincewe assume that these processes are periodic. It does however let us modelsystems like bifurcating structures in cell biology that are non-periodic. Thecorresponding potential is called Waddington's landscape in the biology lit-erature. It is worth noting that this epigenetic landscape in Figure 1.2 is arepresentation in three dimensions which is a very great simplification as thenumber of genes in humans is close to 20,000.As described in the ground truth in Figure 1.1(a), the cells also undergobranching as they develop into more specialized cells. We model this usinga branching Markov process, where each cell moves according to (1.2) andhas a branching rate which depends on time and the position of the cell.51.1. The model and literatureIn summary, we model the entire lineage tree by a branching diffusion withmotion given by (1.2) and branching mechanism given by a suitable functionq.Now, we discuss how to fit the model to the data for inferring the cell devel-opment trajectories. Recall, we need to kill the cell to make the measurementwhich prevents us from observing an individual cells trajectory over time.Thus, as outlined in Figure 1.1(b), it is only possible to obtain independentsamples at different time points from the model. Therefore, one can't usestandard numerical techniques on SDEs to fit the model to the data. In-stead, we employ a different technique using optimal transport.Suppose P is the law of the branching diffusion modelling the lineage tree,one may view the measurements in Figure 1.1(b) as independent samplesfrom the marginal laws Pt at time points t = T1 and t = T2. In Chapter3 we will show that P is uniquely characterized by its temporal marginalsas solution of a relative entropy minimizing problem. Further, from [2], thisvariational problem is equivalent to a regularized unbalanced optimal trans-port problem.Therfore, one may use this optimal transport formulation to construct thelaw P(T1,T2)t for t ∈ [T1, T2] given the marginals PT1 and PT2 at timesT1 and T2 respectively. Furthermore, if we have n observations at times0 = T0 ≤ T1 ≤ · · · ≤ Tn ≤ 1, we can construct the laws P (Ti,Ti+1)t fort ∈ [Ti, Ti+1] and 0 ≤ i ≤ n− 1. Now, we can approximate the law on [0, 1]by patching together the laws P (Ti,Ti+1) for 0 ≤ i ≤ n − 1. As the numberof time points n → ∞, under certain assumptions one can show that thisprocedure yields the true law. One can also add the regularization parameterto compensate for the overfitting.In [20] the authors demonstrate how optimal transport can be applied toinfer trajectories from a scRNA-seq time-course, without any informationabout cell lineage. This method is called Waddington-OT. It also includesentropic regularization to model the inherent stochasticity in developmentaltrajectories.To account for lineage information, in [7], the authors propose a methodcalled LineageOT, which provides a unified framework for lineage tracing61.1. The model and literatureFigure 1.3: This figure is a representation of the optimal transport maps be-tween the different time points. Here Pt1,t2 is the optimal transport couplingbetween Pˆt1 and Pˆt2 which are the empirical observations at time t1 and t2respectively. The x axis represents time, the y axis represents the state ofthe cell in gene expression space which we assume to be Rd.Figure 1.4: This figure contains the added lineage tree information. Usingthis we may estimate the branching mechanism. This estimate can thenbe used in the branching diffusion model considered in the relative entropyminimization problem. The x axis represents time, the y axis represents thestate of the cell in gene expression space which we assume to be Rd.71.2. Overview of the results and layoutand trajectory inference. They use graphical models and the optimal trans-port discussed above to reconstruct developmental trajectories from timecourses with snapshots of both cell states and lineages.A theoretical framework to construct the law of the trajectories of the cellsin gene expression space without branching from samples of its temporalmarginals, is proposed in [13]. They prove that for a class of diffusion pro-cesses with drift, it is possible to recover the ground truth trajectories fromlimited samples of the temporal marginals at each time-point, and providean efficient algorithm to do so in practice. The method they develop, GlobalWaddington-OT (gWOT), boils down to a smooth convex optimization prob-lem posed globally over all time-points involving entropy-regularized optimaltransport. As in [20], the authors don't model branching but allows for vari-able rates of growth across cells.In [4], the authors discuss a grid-free algorithm to compute the minimumentropy estimator in [13] which recovers the dynamics of the drift-diffusionprocesses. They show that the minimizing problem on laws on paths givenmarginals at T time points is equivalent to a reduced minimizing problem onthe laws on the T fold product gene expression space. Notice that the latterproblem is much simpler to compute since the optimization problem is overa finite dimensional product distribution on gene expression space insteadof the laws on the entire path space. Then the authors show us how to findthe minimizer for this simplified problem using gradient descent. They showthat the minimizer is related to the limit of a mean field Langevin (MFL)dynamics and prove it's global convergence to the desired estimator. Theyapproximate MFL using a noisy gradient descent on the finite dimensionalproduct of the gene expression space.1.2 Overview of the results and layoutIn Chapter 2, we discuss the mathematical preliminaries from optimal trans-port and relative entropy required to formulate the entropy minimizationproblem. We present a variational characterization of the law of diffusionas in [13, Theorem 4.1]. Here, the trajectory of cell development is mod-elled by a diffusion process without branching. We conclude this chapter bydiscussing a method to find a computationally efficient solution to the opti-mization problem on path space. We prove that the minimization problem81.2. Overview of the results and layouton path space is equivalent to a reduced problem on the product distributionon gene expression space as shown in [4, Theorem 3.1].In Chapter 3, we discuss an extension of [13, Theorem 4.1] for the case withbranching. More precisely, we show that for a law with a known branchingmechanism, we can recover the true law on branching paths using the samerelative entropy minimizing technique that was used earlier. We considerall laws with the same marginals that we observe and the known branchingmechanism. In Chapter 3, we prove that among these, the law with theminimum relative entropy with respect to branching Brownian motion is thetrue law. We use the Girsanov transformation from [2] for branching diffu-sion and then proceed along the lines of the proof of [13, Theorem 4.1].9Chapter 2Mathematical PreliminariesIn this chapter, we discuss the mathematical preliminaries of optimal trans-port in Section 2.1 and relative entropy in Section 2.2. In Section 2.3, wediscuss a variational characterization of the law of a diffusion process thatwe can recover from the marginals. Finally, in Section 2.4, we discuss datafitting and how to compute the law estimated in Section 2.3 efficiently.We will generalize the result from Section 2.3 in Chapter 3 for the case withbranching and we will consider computational issues and data fitting dis-cussed in Section 2.4 for processes with branching and deaths in future work.In the following section, we provide a brief introduction to the theory of opti-mal transport. We refer the reader to [17, 22, 23] for a detailed introductionof optimal transport and it's computational aspects.2.1 Optimal TransportIn the 1780s, a French mathematician named Gaspard Monge, proposed away to calculate the most effective strategy for moving large amounts ofdirt from once place to another with the least amount of effort. Suppose wehave mass a1, a2, . . . , an of sand at points x1, x2, . . . , xn in d-dimensional Eu-clidean space, Rd, and we wish to distribute it to points y1, y2, . . . , ym witheach yj having mass bj of sand. Let the cost of moving a unit mass of sandfrom xi to yj be given by some cost function c(xi, yj). Then, the Monge prob-lem is to find the pushforward map T : {x1, x2, . . . , xn} → {y1, y2, . . . , ym}that minimizesn∑i=1c(xi, S(xi))over all S : {x1, x2, . . . , xn} → {y1, y2, . . . , ym} such that∀j ∈ {1, 2, . . . ,m}, bj =∑i:S(xi)=yjai. (2.1)102.1. Optimal TransportWe write the above condition (2.1) compactly as S♯α = β or S is a pushfor-ward of α to β. One can immediately notice that a solution to the Mongeproblem need not exist. For example, if we consider a1 = a2 = 1/2 andx1 = 0, x2 = 1 which we wish to transport to b1 = 1/4, b2 = 3/4 andy1 = 4, y2 = 5, it is clear that there is no possible S satisfying (2.1). Onecan also rephrase the problem. Consider two atomic probability measureson any two spaces X and Y,α =n∑i=1aiδxi ∈ P(X ) and β =m∑j=1bjδyj ∈ P(Y). (2.2)The aim of the Monge problem is to transport the mass from α to β, ina way that minimizes a predetermined cost function, c : {x1, . . . , xn} ×{y1, . . . , ym} → R+. We require the existence of a map T : {x1, . . . , xn} →{y1, . . . , ym} satisfying (2.1), which we write in compact form as T♯α = β,that solvesminS{n∑i=1c(xi, S(xi)) : S satisfies (2.1)}.In general, as we noted earlier, Monge maps may not even exist between anytwo arbitrary discrete measures. This can happen when their weight vectorsare not compatible, which is always the case when the target measure hasmore points than the source measure, n < m.Just before the second world war, Leonid Kantorovich, tasked with the jobof finding the most efficient way to distribute goods from factories to ware-houses and stores, laid the foundation of a new branch of mathematics calledLinear Programming. Now we consider the Kantorovich relaxation where theissue with the Monge problem discussed above problem doesn't arise. TheKantorovich relaxation allows mass to be split and then transported to dif-ferent points. We do this by considering couplings of α and β given byΠ(α, β) :={P ∈ Rn×m+ : P1m = α and P T 1n = β}where 1n is the n-dimensional vector with each entry equal to 1. Note thateach P can be thought of as a joint distribution on X × Y with marginalsα and β. The set of matrices Π(α, β) is bounded and is a convex hull of afinite set of matrices. We observe that if α and β have the same total massthen the set Π(α, β) of couplings is always non-empty. Another observationwe can make is that if P ∈ Π(α, β), we must have P T ∈ Π(β, α). The112.1. Optimal TransportKantorovich problem for two measures α and β as in (2.2) and a cost matrixCn×m where Ci,j denotes the cost of moving a unit of mass from xi to yj isto find a solution ofminP∈Π(a,b)n∑i=1m∑j=1Ci,jPi,j . (2.3)The above problem is a linear optimization over Π(α, β) and hence standardresults in linear programming imply that an optimal solution can be found.Now we consider the continuous analogue of these two problems on metricspaces X and Y. We start with the Monge problem for arbitrary measuresµ and ν on X and Y respectively. Given two probability measures µ and ν,we aim to find a pushforward map T : X → Y, that isν(A) = µ(T−1(A)) for any measurable subset A ⊂ Y (2.4)where T minimizes the quantity∫Xc(x, T (x))dµ(x) (2.5)among all the maps satisfying (2.4). Similar issues that we mentioned inthe discrete case continue to be present in this setting as well for the Mongeproblem, unless we make certain absolutely continuous assumptions about µ(See Theorem 2.1.1 below).Now we consider the continuous analogue of the Kantorovich problem ([10]).Given two probability measures µ and ν on X and Y respectively and a costfunction c : X × Y → [0,+∞] we consider the problemmin{∫X×Yc(x, y) dγ(x, y) |γ ∈ Π(µ, ν)}, (2.6)where Π(µ, ν) is the set of transport plans, i.e.Π(µ, ν) := {γ ∈ P(X × Y) : γ|X = µ and γ|Y = ν} .Note that Π(µ, ν) is a non-empty (since γ = µ× ν ∈ Π(µ, ν)) convex set. IfT is a pushforward map then γ = (id × T )♯µ is a coupling. Thus one mayview the Kantorovich problem as a relaxation of the Monge problem.As noted earlier, the Monge problem may not have a solution for example,we would need existence of at least a map T satisfying (2.4). This will not122.2. Relative entropyhappen if µ = δ0, and ν is not a single Dirac mass. There are conditions onc and X when the Monge problem and Kantorovich problem can be solved.We summarize the conditions where this is true below.Theorem 2.1.1. Let X be a compact convex subset of Rd and µ, ν be twoprobability measures on X . Let c : X×X → [0,∞] be a lower semicontinuousBorel cost function then there exists a γ ∈ Π(µ, ν) that solves the Kantorovichproblem (2.6). Further if c is continuous and µ has no atoms, then thereexists a pushforward map T : X → X that solves the Monge problem (2.5)and γ = (id× T )#µ in Π(µ, ν) solves the Kantorovich problem (2.6).For a proof of the above theorem, we refer the reader to [1, Theorem 2.1]and for a detailed discussion of the two problems, [22, Chapter 4]. Theminimizers for the Kantorovich problem above are called optimal transportplans between µ and ν.2.2 Relative entropyIn this section, we discuss the concept of entropic cost, and the connectionbetween relative entropy and regularized optimal transport. We begin witha definition of entropy for discrete and continuous measures.Let X be a convex subset of Rd, and Y be a discrete random variable on Xwith probability mass function p on X , we define its entropy asH(p) =∑x∈Range(Y )p(x) log p(x) = E[log p(Y )].If we let Z be a continuous random variable on X with probability densityfunction f on X , we define its entropy asH(f) =∫Xf(x) log f(x)dx = E[log f(Z)].Note that the above quantities are negative and will be assumed to have thevalue −∞ if the series or the integral diverges. This definition of entropy isthe negative of the usual Shannon differential entropy.We will now discuss the regularized optimal transport problem connectedwith the Kantorovich problem. First, the discrete entropy of a probabilitymatrix P = (Pi,j) is defined asH(P ) =: −∑i,jPi,j(log(Pi,j)− 1),132.2. Relative entropywith the convention that H(P ) = −∞ if one of the entries Pi,j is 0. The ideaof regularized entropy of optimal transport mimics the idea of regulariza-tion to find approximate solutions of optimization problems. The ε-entropyregularized optimal transport problem for (2.6) is phrased as followsminP∈Π(α,β)∑i,jCi,jPi,j + εH(P ), (2.7)for any ε > 0. Since the objective function is convex, (2.7) has a uniqueoptimal solution. As ε → 0, the unique optimal solution Pε converges toP , the solution of the Kantorovich problem (2.3) (See [17, Proposition 4.1]).The above entropy regularization problem is used in several computationalalgorithms as well (See [5, 17]). The complexity of the Kantorovich problemdiffers from that of entropy regularized optimal transport. More precisely,the Sinkhorn algorithm finds a minimizer of (2.7) in order of n2 log n stepswhereas a standard algorithm to solve the Kantorovich problem will takeorder of n3 steps. We refer the reader to a detailed discussion on such com-putational aspects in [17, Section 4.2].Now, we introduce relative entropy whose minimization is related to entropyregularized optimal transport. Suppose we have two probability measures µand ν on X then, the Kullback-Leibler distance or relative entropy of µ withrespect to ν is defined asH(µ | ν) ={ ∫µ(dy) log dµdν (y), if µ≪ ν,∞, otherwise.In this definition, we use the convention that 0 log 0 = 0. The relative en-tropy of µ with respect to ν is non-negative. In statistics, it arises as anexpected logarithm of the likelihood ratio.We are now going to introduce the continuous analogue of entropy regularizedoptimal transport and discuss how it relates to relative entropy. Let X = Y =Rd and consider two probability densities f and h on Rd and an integrablecost function c : Rd×Rd → [0,∞). The corresponding ε-entropy regularizedoptimal transport problem in the continuous setting is to find a solution tominγ∈Π(f,h)∫Rd×Rdc(x, y)dγ(x, y) + εH(γ) (2.8)for ε > 0. Let gε : Rd × Rd → [0,∞) be the joint density given bygε(x, y) = f(x)1cεexp(−c(x, y)ε)142.3. A variational characterization of the law of a diffusionwhere cε is a normalizing constant. Let γ ∈ Π(f, h) be any coupling suchthat γ ≪ gε. Note that the relative entropy of γ with respect to gε can bewritten asH(γ | gε) =∫dγ(x, y) log(dγ(x, y)dgε(x, y))= H(γ)−∫dγ(x, y) log(gε(x, y))= H(γ)−∫dγ(x, y) log(f(x)) +1ε∫dγ(x, y)c(x, y)− log(cε)= H(γ)−H(f) + 1ε∫dγ(x, y)c(x, y)− log(cε)=1ε(∫dγ(x, y)c(x, y) + εH(γ))−H(f)− log(cε).Since f and cε are given to us, the minimizing problem (2.8) is equivalent tominimizing the relative entropy that is,minγ∈Π(f,g)H(γ | gε). (2.9)This equivalence will be useful in characterizing many optimization problems.In the above calculation, we consider X = Y = Rd, the connection between(2.8) and (2.9) holds in much more general settings (See [4, 17]).2.3 A variational characterization of the law of adiffusionIn this section, we discuss a key result from [13], that provides a theoreti-cal framework to understand trajectory inference for cell development if itis modelled by a diffusion in gene expression space without any branching.More precisely, the result [13, Theorem 4.1] states that the law of the SDE(1.2) can be determined by minimizing relative entropy with respect to Brow-nian motion given the marginals. We will state and prove the same resultwhen we assume the diffusion in (1.2) to be on Td.Let Td be the d dimensional torus. It is a compact smooth d dimensionalRiemannian manifold without boundary and let the Laplace-Beltrami oper-ator on C2(Td) be denoted by ∆. The normalized volume measure on Tdis given by vol such that∫Td dvol = 1. Let Ω = C([0, 1],Td) denote the set152.3. A variational characterization of the law of a diffusionof continuous paths on Td and P(Ω) be the space of all probability mea-sures on Ω. We denote by (Xt)t∈[0,1] the canonical process on Ω, that is foreach t ∈ [0, 1], Xt(ω) = ωt. If R ∈ P(Ω) is a probability measure on thespace of paths Ω, we denote by Rt ∈ P(Td) its marginal at time t. Thatis, if X is a random element of Ω distributed according to R then Xt isdistributed according to Rt. Let Ft denote the Borel σ-algebra generated bythe random variablesXs for s ≤ t, in such a way that (Ft)t∈[0,1] is a filtration.By a Wiener measure with diffusivity σ2, we mean an element of P(Ω)which is a diffusion measure generated by the second order elliptic opera-tor f 7→ σ22 ∆f (in the sense of [8, Definition 1.3.1]). Let {Bσt }t∈[0,1], the ddimensional Brownian motion with diffusivity σ2, be the stochastic processwhose law corresponds to the Weiner measure with diffusivity σ2. By thelaw of solutions of the SDE (1.2), we mean diffusion measure generated byf 7→ σ22 ∆f − ∇Ψ · ∇f . We denote by Bσ the law of a Brownian motionstarting with initial measure given by vol. This ensures that the Brownianmotion under Bσ is reversible. Now we are ready to state the main result ofthe section, which is [13, Theorem 2.1].Theorem 2.3.1. Let P0 ∈ P(Td) be a probability distribution withH(P0|vol) < +∞. Let Ψ : [0, 1] × Td → R be a time dependent potential inC2([0, 1]× Td). We consider P the law of the SDEdXt = −∇Ψ(t,Xt)dt+ dBσt (2.10)with initial condition P0. Then, if R ∈ P(Ω) is such that Rt = Pt for allt ∈ [0, 1], there holdsH(P |Bσ) ≤ H(R|Bσ)with equality if and only if P = R.We will follow the proof as in [13, Theorem 4.1]. We now state two results([13, Proposition 2.10 and 2.11]) whose proofs we will postpone, assumingwhich we prove the theorem.Proposition 2.3.2. Let B˜σ ∈ P(Ω) denote a Wiener measure whose initialdistribution is not necessarily vol. Let φ : [0, 1] × Td → R be a smooth162.3. A variational characterization of the law of a diffusionfunction. Then, the process whose value at time t ∈ [0, 1] is given byexp(1σ2(φ(t,Xt)− φ(0, X0)−∫ t0[∂sφ+12|∇φ|2 + σ22∆φ](s,Xs)ds))(2.11)is a Ft-martingale under B˜σ.Let P be as in Theorem 2.3.1. We will now state a proposition which de-scribes the Radon-Nikodym derivative of P with respect to Bσ. In [13], theauthors prove the result without using stochastic integration over the mani-fold and only use the martingale characterizations.Proposition 2.3.3. In the framework above, the Radon-Nikodym derivativeof P with respect to Bσ is given Bσ-a.e. bydPdBσ(X) =dP0dvol(X0) exp(1σ2(Ψ(0, X0)−Ψ(1, X1)+∫ 10(∂sΨ− 12|∇Ψ|2 + σ22∆Ψ)(s,Xs)ds)).The key point of the above proposition is that the density of P with respectto Bσ does not involve a stochastic integral. This is because the drift in theSDE is the gradient of a potential.We now present the proof of the main result of this section. As mentionedbefore, the proofs are borrowed from [13].Proof of Theorem 2.3.1. Let P be the law of the solution of the SDE(2.10) and R be another probability distribution on Ω such that H(R|Bσ) <+∞ (otherwise the result trivially holds). Let p, r respectively denote theintegrable Radon-Nikodym derivative of P and R with respect to Bσ. Bystrict convexity of the function x 7→ x log x,r log r − p log p ⩾ (1 + log p)(r − p),holds Bσ a.e. with equality if and only if r = p. By integrating with respectto Bσ, we find thatH(R|Bσ)−H(P |Bσ) ⩾ ER [1 + log p]− EP [1 + log p] . (2.12)172.3. A variational characterization of the law of a diffusionOn the other hand, given Proposition 2.3.3, we haveER [1 + log p] = ER[1 + log(dP0dvol)(X0)+1σ2(Ψ(0, X0)−Ψ(1, X1) +∫ 10(∂sΨ− 12|∇Ψ|2 + σ22∆Ψ)(s,Xs)ds)].As this expression depends only on the temporal marginals of R, it showsthat the right hand side of (2.12) vanishes if Rt = Pt for all t ∈ [0, 1]. Thiscompletes the proof.To complete the proof of the theorem, we now prove Proposition 2.3.3 andProposition 2.3.2.Proof of Proposition 2.3.2. With respect to B˜σ, the following stochastic pro-cessLφt = φ(t,Xt)− φ(0, X0)−∫ t0[∂sφ+σ22∆φ](s,Xs)dsis a bounded martingale and its quadratic variation is given by (See [9,Section 1.3])σ2∫ t0|∇φ(s,Xs)|2ds.Then (2.11) is nothing else than the exponential martingale [14, Proposition5.11] associated to σ−2Lφ.Proof of Proposition 2.3.3. For each x ∈ Td, let Bσ,x be the law of the Brow-nian motion starting from x, that is such that Bσ,x0 = δx. Define a processD whose value at time t is given byDt =exp(1σ2(Ψ(0, X0)−Ψ(t,Xt)+∫ t0(∂sΨ− 12|∇Ψ|2 + σ22∆Ψ)(s,Xs)ds)).As stated in Proposition 2.3.2, under Bσ,x this is a bounded martingale: itis the exponential martingale of the process (Nt)t∈[0,1], defined byNt =1σ2(Ψ(0, X0)−Ψ(t,Xt) +∫ t0(∂sΨ+σ22∆Ψ)(s,Xs)ds).182.4. Trajectory inference using diffusion modelswhose quadratic variation is σ−2∫ 10 |∇Ψ(s,Xs)|2ds. As a consequence, wecan define P˜ x = D1Bσ,x ∈ P(Ω). Then, let us take f ∈ C∞(Td,R). UnderBσ,x, we know that the process Lf whose value at time t is given byLft = f(Xt)− f(X0)−∫ t0σ22∆f(Xs)dsis a martingale with quadratic variation given by σ2∫ t0 |∇f(Xs)|2ds. Apply-ing Girsanov's theorem [14, Theorem 5.22] for real-valued semi martingales,we know that under P˜ x the process(f(Xt)− f(X0)−∫ t0σ22∆f(Xs)ds+∫ t0∇f(Xs) · ∇Ψ(Xs)ds)t∈[0,1]is a local martingale. As it is clearly bounded, it is a martingale. This exactlyshows that P˜ x is a diffusion measure generated by f 7→ σ22 ∆f − ∇Ψ · ∇f ,and its initial distribution is Bσ,x0 = δx. Eventually we average in x: we candefineP˜ =∫TdP˜ x P0(dx) =dP0dvol(X0)D1Bσ,and by an easy conditioning argument we see that P˜ is still a measure gen-erated by the diffusion corresponding to the generator f 7→ σ22 ∆f −∇Ψ ·∇fwith initial distribution P0. By uniqueness we have, P˜ = P .2.4 Trajectory inference using diffusion modelsIn Theorem 2.3.1, we established that the law on paths P of the stochasticdifferential equation (2.10) can be recovered from the marginals by mini-mizing the relative entropy. In the example in Figure 1.1, we discussed thesnapshots obtained by scRNA-sequencing and in Figure 1.3 and 1.4, we dis-cussed how we would use optimal transport to estimate the ground truth P .In this section, we explain how to solve the optimization problem efficientlygiven the snapshots. We will follow [4, 13] to present a broad overview ofthe method.We will begin by discussing the minimizing problem on path space. Let{µˆti}Ti=1 be the observed marginals from scRNA-sequencing. In [13], theypropose the Global Waddington-OT (gWOT) method that solves the datafitting problem by finding the minimizer of F : P(Ω) → R which is of the192.4. Trajectory inference using diffusion modelsformF(R) = Fit(Rt1 , . . . , RtT ) + σH(R|Bσ) (2.13)where Fit is a convex function that depends on the temporal marginals ofR at the measurement times t1, . . . , tT . We note that the above problemon P(Ω) is in general hard to solve efficiently, so in [4], the authors showequivalence of the above problem to one on a reduced space P(Td)T .We now discuss the reduced problem. For µ, ν ∈ P(Td), the relative entropycost between µ and ν is defined, for some τ > 0, asTτ (µ, ν) = minγ∈Π(µ,ν)τH(γ|pτµ⊗ ν) (2.14)where pt(x, y) is the transition probability density of the (reflected) Brownianmotion on Td over the time interval [0, t]  or equivalently the heat kernelwith boundary conditions.Consider the function G : P(Td)T → R defined for µ = (µ(1), . . . , µ(T )) thatrepresents a family of T reconstructed temporal marginals, byG(µ) := Fit(µ) +T−1∑i=11ti+1 − tiTσi(µ(i), µ(i+1))where σi := (ti+1 − ti)σ. We now introduce the reduced objective functionF : P(Td)T → R, defined asF (µ) := G(µ) + σH(µ). (2.15)Remark 2.4.1. The following Fitη function is considered in [4],Fitη(Rt, µˆt) :=∫− log[∫exp(− ∥x− y∥22η2)dRt(x)]dµˆt(y)= H(µˆt|Rt ∗ gη) +H(µˆt) + C (2.16)where gη(x) = (2πη2)−d2 e−∥x∥2/(2η2) is the Gaussian density and C > 0is a constant. This is the negative log-likelihood under the noisy observa-tion model Xˆti,j = Xti,j + ηZi,j where Xˆti,j is the observation and Zi,jiid∼Normal(0, I) the noise. Suppose Fit(Rt1 , . . . , RtT ) is taken to beFit(Rt1 , . . . , RtT ) :=1λT∑i=1∆tiFitη(Rti |µˆti)202.4. Trajectory inference using diffusion modelswhere λ > 0 is the regularization strength, ∆ti := (ti+1 − ti−1)/2 (with theconvention t−1 = 0 and tT+1 = 1) and Fitη (2.16) is a divergence functionalthat quantifies how much Rti and µˆti differ. Then, it is shown that in [13,Theorem 2.3] if (ti)i∈[T ] becomes dense in [0, 1] as T grows,limλ,η→0limT→∞R∗ = P weakly, almost surely,where R∗ is the minimizer of (2.13). Thus (2.13) consists of a convex Fitand the relative entropy with respect to Bσ but however the space P (Ω) makesit difficult to design efficient algorithms. In [4] the authors propose the abovereduction of the minimizing problem of F on P (Ω) to minimizing F onP(Td)T . They solve this reduced problem using the Mean-Field Langevin(MFL) method. See [4, Section 3.2-3.4] for the reduced problem, its globalconvergence guarantees and also a discretisation method which employs astochastic gradient descent to obtain the MFL dynamics in the mean-field,vanishing step-size limit.Here, we discuss the equivalence result shown in [4].Theorem 2.4.2 (Representer Theorem). Let Fit : P(Td)T → R be anyfunction.1. If F admits a minimizer R∗ then (R∗t1 , . . . , R∗tT ) is a minimizer for F .2. Conversely, if F admits a minimizer µ∗ ∈ P(Td)T then a minimizerR∗ for F is built asR∗(·) =∫(Td)TBσ(·|x1, . . . , xT )dRt1,...,tT (x1, . . . , xT )where Bσ(·|x1, . . . , xT ) is the law of Bσ conditioned on passing throughx1, . . . , xT at times t1, . . . , tT respectively and Rt1,...,tT is the composi-tion of the transport plans γi,i+1 which are optimal in the definition ofTσi(µ∗(i), µ∗(i+1)), for i = 1, . . . , T − 1.We note that the first optimization problem (2.13) is over the laws on pathson Td, which is a much larger space compared to laws on the product space(Td)T , which is what the second optimization problem (2.15) is over. Thusthe above result offers a significant complexity reduction. Further, note thatTheorem 2.4.2 provides a method to construct a minimizer for F from aminimizer for F and the associated optimal transport plans (γi,i+1)1≤i≤T−1.212.4. Trajectory inference using diffusion modelsWe conclude this section by providing proof of Theorem 2.4.2. This proof istaken from the proof of [4, Theorem 3.1]Proof of Theorem 2.4.2. We provide only the key details of the proof of the-orem below assuming that Fit is convex and refer the reader to [4, LemmaB.1] for the complete proof. The broad idea is to show that there is a bi-jection between the two problems. We begin with a characterisation of theminimizers of (2.13). Suppose the (unique) minimizer R ∈ P(Ω) of (2.13)has marginals Rt1 , . . . , RtT ∈ P(Td) at instants t1, . . . , tT respectively. Fur-thermore, let Rti,ti+1 ∈ Π(Rti , Rti+1) be the optimal coupling from (2.14)corresponding to Tσ∆ti(Rti , Rti+1) for each i. Then in [13, Proposition 3.2]it is shown thatRt1,...,tT = Rt1,t2 ◦ · · · ◦RtT−1,tTis the unique Markov process whose pairwise couplings are Rti,ti+1 (2.17)and the law R that minimizes (2.13) can be fully characterized in terms ofthe finite-dimensional distribution (2.17) byR(·) =∫(Td)TBσ(·|x1, . . . , xT )dRt1,...,tT (x1, . . . , xT ) (2.18)where Bσ(·|x1, . . . , xT ) is the law of the reversible Brownian motion withdiffusivity σ2 conditioned on passing through x1, . . . , xT at times t1, . . . , tTrespectively.For the rest of the proof we assume that R satisfies (2.17) and (2.18). Fromthe additivity property of the relative entropy under conditioning we haveH(R|Bσ) =H(Rt1,...,tT |Bσt1,...,tT )+∫H(R(·|x1, . . . , xT )|Bσ(·|x1, . . . , xT ))dRt1,...,tT (x1, . . . , xT )where the second term vanishes if and only if the conditional distributionsR(·|x1, . . . , xT ) are Brownian bridges, for Rt1,...,tT almost every (x1, . . . , xT ).Thus, if R satisfies (2.18)H(R|Bσ) = H(Rt1,...,tT |Bσt1,...,tT ). (2.19)Secondly,H(Rt1,...,tT |Bσt1,...,tT ) =T−1∑i=1H(Rti,ti+1 |Bσti,ti+1)−T−1∑i=2H(Rti |Bσti). (2.20)222.4. Trajectory inference using diffusion modelsWe will examine each term in above separately. Without loss of generality,assume that Rti are absolutely continuous with densitydRtidx (x) = ri(x) andby a slight abuse of notation we will let vol be the Lebesgue volume of Td.On the one hand, since Bσti is the normalized volume measure on Td, it holdsH(Rti |Bσti) = H(Rti) + log(vol) and Bσti,tt+1(dx, dy) = vol−1pσi(x, y)dxdy(2.21)by definition of the transition probability density p of the Brownian motionon Td. Secondly, it follows that for any µ, ν ∈ P(Td) with finite differentialentropy and γ ∈ Π(µ, ν) it holdsH(γ|Bσti,ti+1) =∫log( dγdx⊗ dyvolpσi)dγ(x, y)= log(vol) +∫log( dγpσidµ⊗ νdµdxdνdy)dγ= log(vol) +H(γ|pσiµ⊗ ν) +H(µ) +H(ν). (2.22)Now using (2.21) and (2.22) with Rti,ti+1 instead of γ in (2.20) we haveT−1∑i=1H(Rti,ti+1 |Bσti,ti+1)−T−1∑i=2H(Rti |Bσti)= log(vol) +T−1∑i=1H(Rti,ti+1 |pσiRti ⊗Rti+1) +T−1∑i=1H(Rti) +T∑i=2H(Rti)−T−1∑i=2H(Rti)= log(vol) +T−1∑i=1H(Rti,ti+1 |pσiRti ⊗Rti+1) +T∑i=1H(Rti).Using the above in (2.19), we haveH(R | Bσ) = H(Rt1,...,tT |Bσt1,...,tT )=T−1∑i=1H(Rti,ti+1 |pσi(Rti ⊗Rti+1)) +T∑i=1H(Rti) + log(vol).(2.23)To conclude the proof, we will use (2.23) and the proof outlined in [4,Theorem 3.1]. As indicated earlier, a minimizer R∗ ∈ P(Ω) of F(R) =232.4. Trajectory inference using diffusion modelsFit(R1, . . . , RT ) + σH(R|Bσ) satisfies (2.17) and (2.18) and hence (2.23)holds. Let µ(i) = R∗ti be its marginals and γ(i) = R∗ti,ti+1 which clearlysatisfies γ(i) ∈ Π(µ(i), µ(i+1)). It holds, with C = log(vol),F(R∗) = Fit(µ(1), . . . , µ(T )) + σT−1∑i=1H(γ(i)|pσiµ(i) ⊗ µ(i+1)) + σH(µ) + C≥ Fit(µ(1), . . . , µ(T )) + σσiT−1∑i=1Tσi(µ(i), µ(i+1) + σH(µ) + Cwhere the last equality holds if and only if R∗ti,ti+1 = γ(i)is optimal inthe definition of Tσi(µ(i), µ(i+1). Thus, the bijection holds and the proof iscomplete.24Chapter 3A variational characterizationof the law of a branchingdiffusionSo far in Chapter 2, Section 2.3, we have discussed a way to model the move-ment of cells in gene expression space with a diffusion process. Recall thatfrom scRNA sequencing, we are able to find time marginals of independentsamples of cells at different time points and using CRISPR we are able toestimate a lineage tree. In this chapter, we now aim to use this informationand estimate the law of the paths that the cells take by minimizing the rel-ative entropy with respect to branching Brownian motion.We assume that each cell, independent of all other cells, lives for an expo-nential lifetime during which it moves according to the gradient of a smoothtime dependent potential. Upon it's death, it produces a random number ofoffsprings which move independently. We show that we can characterize thelaw of the branching diffusion given it's marginals by a variational probleminvolving relative entropy. More precisely, suppose P is the true law of thebranching diffusion then we show that it has the least relative entropy withrespect to a given branching Brownian motion among all the laws with thesame marginals.We begin with the setup where we precisely define branching diffusion, andshow that using a result from [2], we can construct it as an absolutely con-tinuous measure with respect to a given branching Brownian motion with abranching mechanism that's constant in space and time. Then we state andprove our main result, Theorem 3.2.1. We conclude the chapter by discussinglimitations and possible future work.253.1. Setup3.1 SetupIn this section we shall describe precisely the Branching diffusion processwhich is used to model cell-development trajectory. Each cell will live an ex-ponential amount of time and upon its death gives rise to a random numberof cells. During its lifetime the cell will perform a diffusion in gene expressionspace.We will first define branching diffusion as the solution to a martingale prob-lem. To understand this we begin by describing two parameters, namely, thebranching mechanism and the motion.Branching Mechanism: We assume that each cell in the system, has life-time determined by an exponential random variable with parameter λ(t, x)which depends on both time t and location of the cell x in space. We refer toλ as the branching rate of the cell. When the clock for a cell located at x attime t rings, it dies and is replaced by k ≥ 1 cells with probability pk(t, x).We refer top(t, x) = (pk(t, x))k∈N, with pk(t, x) ≥ 0,∑k≥1pk(t, x) = 1as the offspring distribution. We denote the branching mechanism of theprocess by q whereq˜(t, x) = {q˜k(t, x) : k ∈ N, t ∈ [0, 1], x ∈ Td} (3.1)where q˜k(t, x) = λ(t, x)pk(t, x). Notice that the branching rate λq˜(t,x) =λ(t, x) =∑k≥1 q˜k(t, x). For a cell at position x at time t, q˜k(t, x) denotesthe rate with which the cell dies and is replaced by k new cells. Given ourmotivation of modelling the trajectory of cell development, we always assumethat p1 = q˜1 = 0, since we are not interested in branching events where acell is replaced by exactly one cell. We assume that q˜ is a continuous nonzero function that has finite first order moment i.e., there exists c1 > 0 suchthatq˜(·, ·) ̸≡ 0 is continuous and∑k≥1kq˜k(·, ·) < c1. (3.2)Given (3.2), for any t ≥ 0 there are finitely many branching events that havehappened until time t and the expected number of cells alive at time t is alsofinite.263.1. SetupFor each (t, x) let the generating function of q˜ be denoted by Φq˜(t,x) : R+ 7→[0,∞) be given byΦq˜(t,x)(z) =∑k≥1q˜k(t, x)zk.We abuse notation and use q˜ and q˜(t) when referring to q˜(t, x).We now assume the existence of an abstract probability space (Ω,F ,P) suchthat all the random variables below are defined on it. Let E be the expecta-tion with respect to P.Cell Motion on Td: Let Td be a d dimensional torus. It is well knownthat Td is a compact Reimann manifold without boundary. Let C2(Td) bethe space of continuously twice differentiable functions on Td and let theLaplace Beltrami operator on C2(Td) be given by ∆. Let vol be the normal-ized volume measure on Td.Let (Bt)t≥0 denote the Brownian motion corresponding to the Laplace Bel-trami operator ∆ on Td with B0 = 0. Each cell will start at an arbitrarypoint y0 ∈ Td and follow trajectory for t ∈ [0, 1] given bydYt = v(t, Yt)dt+ σdBt (3.3)where v : R+ × Td → Td is a continuous function, σ > 0 and Y0 = y0.A finite measure, Mt, on Td is called a branching diffusion if1. P a.s,∫ 10⟨|v(t)|,Mt⟩ dt < +∞ and∫ 10⟨λ(t),Mt⟩ dt <∞, and2. for all smooth ϕ : [0, 1]× Td → R+, the processexp(⟨log(ϕ(t),Mt⟩ − ⟨log ϕ(0),M0⟩ −∫ t0〈 1ϕ(∂tϕ(s)) ,Ms〉ds−∫ t0〈 1ϕ(σ2∆ϕ(s) + v(s) · ∇ϕ(s)Φq˜,t,x(ϕ(s))− λϕ(s)),Ms⟩ds)is a local martingale under P.273.1. SetupWe shall denote the marginals of the Branching diffusion by the finite mea-sure, Mt, on Td. We now follow [2] and describe a construction of the aboveBranching diffusion process as a change of measure of branching Brown-ian motion with constant branching rates. We begin by defining branchingBrownian motion.Branching Brownian motion R ∼ BBM(σ, q,R0): As stated abovewe shall assume each cell's motion is governed by (3.3) with v ≡ 0 and thebranching mechanism (3.1) is constant over space-time, i.e. q = {qk : k ∈ N}and satisfies (3.1). Let λq =∑k≥1 qk > 0 and p =qλ To keep track of thegeneology of the branching system, we introduce a labelling scheme for thecells. We use the set U := ∪k∈NNk of finite sequences of natural numbersto label the cells. Every cell in the system is given a label from U and thecell labelled n1 . . . nk−1nk ∈ U to be the nk-th descendant of the cell labelledn1 . . . nk−1 ∈ U . Further, we introduce a partial ordering of U by sayingn′ < n if n′ = n′1 . . . n′l and n = n1 . . . nk are such that l < k and n′j = nj for1 ≤ j ≤ l. This is the same as saying: n′ < n iff the cell n′ is an ancestor ofn in the branching system. Let (Ln, en, Yn)n∈U be a sequence of independentrandom variables in Ω where for all n ∈ U , we have Ln ∈ N denotes the number of descendants of the cell n. It is drawnfrom the law p = (pk)k∈N ∈ P(N). en ∈ R denotes the lifetime of cell n. It follows an exponential lawwith parameter λq =∑k qk. Yn ∈ C(R+,Td) i.i.d. Y denotes the trajectory of the cell labelled n.Suppose we start with cells labelled {1, . . . , p} at locations {x1, . . . , xp}. Wedefine T recursively by T0 := {1, . . . , p} and for all k ≥ 0:Tk+1 :={nj ∈ U : n ∈ Tk, j ∈ {1, . . . , Ln}},and finally T := ⋃k∈N Tk. Note that Tk denotes the k-th generation of cellsfor all the trees. For any cell n ∈ T , let bn and dn denote the time of birthand death of the two cells respectively. We havebn =∑n′<nen′ and dn = bn + en.The time of birth of the n-th cell is simply the sum of the lifetimes of allit's ancestors. We denote the set of cells alive at time t ≥ 0 with N (t) =283.1. Setup{n ∈ T : bn ≤ t < dn} . For a given time t ∈ [bn, dn) and n ∈ N (t), theposition of the cell n at time t is given byXn(t) = xn1 +∑n′<nYn′(en′) + Yn(t− bn).For t ≥ 0, we define the empirical measure associated with the positions ofthe cells alive at time t by Mt ∈M(Td) whereMt =∑n∈N (t)δXn(t).The measure valued curve M = (Mt)t∈[0,1] is almost surely cádlág for thetopology of weak convergence. Note that the law of Mt depends on µ =δx1 + · · · + δxp , and not on the labelling of x1, . . . , xp. (See [2, Proposition2.29]).We shall refer to R as the law of Branching Brownian motion with diffusivityσ > 0 and branching mechanism q with initial law R0. We define verbatimfrom [2], two objects from branching Brownian motion. The first one willbe a continuous martingale that is a direct generalization of the stochasticintegral with respect to the Brownian motion:I[f ]t =∑n∈U∫ t01{bn<s≤dn}f(s,Xn(s))dXn(s), (3.4)where f : R+×Td → R is any continuous function and bn, dn, Xn are definedabove. it consists in adding all the stochastic integral on all the branches ofthe BBM. The second one will be a simple pure-jump process whose jumpinstants will be the branching times, and whose jump sizes will depend onthe time of the branching event, the position of the branching event, thenumber of descendants of the branching events, and the past of the wholeprocess. Namely1,J k [f ]t =∑s≤t1{∆Ms(Td)=k−1}〈f(s),∆Msk − 1〉, (3.5)where f : R+×Td → R is any continuous function and bn, dn, Xn are definedabove. The following generalised Ito formula is shown in [2, Theorem 4.19]1For any g : Td → R and any measure µ on Td, let ⟨g, µ⟩ denote the integral of g withrespect to µ. That is ⟨g, µ⟩ = ∫Td g(x)dµ(x).293.1. SetupTheorem 3.1.1. Let φ : [0, 1] × Td → R be a smooth function. We haveP-a.s. for all t ∈ [0, 1]:⟨φ(t),Mt⟩ =⟨φ(0),M0⟩+ I[∇φ]t ++∞∑k=1J k[(k − 1)φ]t+∫ t0〈∂tφ(s) +σ2∆φ(s),Ms〉ds.We use the two terms I and J in the construction below.Construction of a branching diffusion: In [2, Theorem 4.23], they con-struct Branching difussion as the Radon-Nikodym derivatives w.r.t. Branch-ing Brownian motion. Let q˜ be as in (3.1). We will assume there is a θ suchthat for all (t, x) ∑k≥1exp(kθ)q˜k(t, x) <∞ (3.6)and relative entropy h : [0, 1]× Td given byh(q˜|q)(t, x) :=∑k≥1q˜k(t, x) logq˜k(t, x)qk+ qk − q˜k(t, x) (3.7)is a bounded function.The following result is contained in [2, Theorem 4.23].Theorem 3.1.2. Let R ∼ BBM(σ, q,R0), P0 ∈ P(M(Td)) with P0 ≪ R0and h, q are as above. The measure P whose Radon-Nikodym derivative isgiven bydPdR:=dP0dR0(M0) exp(1σI[∇Ψ]1 +∑kJ k[logq˜kqk]1−∫ 10〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,Mt〉dt)(with convention log 0/0 = 0) is the (unique) law of the branching diffusionwith motion given by (3.8) with v = −∇Ψ and branching mechanism givenby (3.1).303.2. Main result3.2 Main resultIn this section, we state and prove our main result. We show that the true lawcan be identified as the relative entropy minimizer with respect to branchingBrownian motion.Theorem 3.2.1. Let R ∼ BBM(σ, q,R0) be as above. Let P0 ∈ P(Td) be aprobability measure such that H(P0 | R0) <∞. Let q˜ be as in (3.1) satisfying(3.2), (3.7) and (3.6). Let Ψ : [0, 1]× Td → R be a time dependant potentialin C2([0, 1] × Td). Finally, Let P ∈ P(Ω) be the law of branching diffusionwith cell motion given bydYt = −∇Ψ(t, Yt) + σdBt (3.8)and branching mechanism (q˜k)k≥1 and initial condition P0. If Q ∈ P(Ω) isany other law of a branching diffusion with the same branching mechanism(q˜k)k≥1 such that Pt = Qt for all t ∈ [0, 1] then the following holdsH(P | R) ≤ H(Q | R) (3.9)where R ∼ BBM(σ, q,R0) with equality iff P = Q.Proof. Let P , Q and R be as in the statement of the theorem. Without lossof generality, we may assume that H(Q | R) <∞. From Theorem 3.1.2, weknow the exact expression fordPdR is given bydPdR=dP0dR0(M0) exp 1σI[∇Ψ]1 +∑k≥1J k[logq˜kqk]1−∫ 10〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,Mt〉dt)(3.10)where I and J are defined in the above section in (3.4) and (3.5) respectively.Now, observe that x 7→ x log x is convex. So,dQdRlog(dQdR)− dPdRlog(dPdR)≥(1 + log(dPdR))(dQdR− dPdR).Integrating the two sides of this equation with respect to R we get313.2. Main resultH(Q | R)−H(P | R) ≥ EQ[1 + logdPdR]− EP[1 + logdPdR]. (3.11)We want to evaluate the right side of (3.11) using (3.10).EQ[1 + log(dPdR)]− EP[1 + log(dPdR)]= EQ1 + log( dP0dR0(M0))+1σI[∇Ψ]1 +∑k≥1J k[log(q˜kqk)]1−∫ 10〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,Mt〉dt]− EP1 + log( dP0dR0(M0))+1σI[∇Ψ]1 +∑k≥1J k[log(q˜kqk)]1−∫ 10〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,Mt〉dt]. (3.12)Now we make a couple of observations. First,EQ[1 + log(dP0dR0(M0))]− EP[1 + log(dP0dR0(M0))]= 0, (3.13)since these two terms only depend on the marginals and both P and Q havethe same marginals. Second, it is well known that for every t ∈ [0, 1], underassumptions on q˜ and q (See [2, Proposition 2.56]), there exists K > 0 suchthatEQ [⟨1,Mt⟩] = EP [⟨1,Mt⟩] < K. (3.14)Further, using the above along with the assumption |∇Ψ(·, ·)| is boundedand since q˜ is uniformly bounded and q is constant, by Fubini's theorem wehaveEQ[∫ 10〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,Mt〉dt]=∫ 10EQ[〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,Mt〉]dt=∫ 10〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,EQ [Mt]〉dt, (3.15)323.2. Main resultwhere the second step uses the definition of mean measure. Using the sameargument for P , we getEP[∫ 10〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,Mt〉dt]=∫ 10〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,EP [Mt]〉dt. (3.16)Now, since the marginals of P andQ are the same, we have EP [Mt] = EQ[Mt]for all t ∈ [0, 1]. From (3.16) and (3.15), we have0 =EQ[∫ 10〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,Mt〉dt]− EP[∫ 10〈 |∇Ψ(t)|22σ+ λq˜(t) − λq,Mt〉dt]. (3.17)Applying (3.13) and (3.17) to (3.12) we haveEQ[1 + log(dPdR)]− EP[1 + log(dPdR)]=(EQ[1σI[∇Ψ]1]− EP[1σI[∇Ψ]1])+EQ∑k≥1J k[log(q˜kqk)]1− EP∑k≥1J k[log(q˜kqk)]1:= (I) + (II). (3.18)We analyse (I). From Theorem 3.1.1, we haveI[∇ϕ]t =⟨ϕ(t),Mt⟩ − ⟨ϕ(0),M0⟩ −∑k≥1J k[(k − 1)ϕ]t−∫ 10〈∂tϕ(s) +σ2∆ϕ(s),Ms〉ds.We can substitute the above with ϕ = Ψ at t = 1 into (I) in (3.18) to obtain333.2. Main result(I) =EQ⟨Ψ(1),M1⟩ − ⟨Ψ(0),M0⟩ −∑k≥1J k[(k − 1)Ψ]1−∫ 10〈∂tΨ(s) +σ2∆Ψ(s),Ms〉ds]− EP⟨Ψ(1),M1⟩ − ⟨Ψ(0),M0⟩ −∑k≥1J k[(k − 1)Ψ]1−∫ 10〈∂tΨ(s) +σ2∆Ψ(s),Ms〉ds].Since we assume that Ψ ∈ C2([0, 1] × Td), ∂tΨ(·, ·) + σ2∆Ψ(·, ·) is bounded.Now we use a similar Fubini argument, definition of mean measure as in(3.13)  (3.17) and the fact that the marginals match (i.e., EP [Mt] = EQ[Mt])to obtain(I) = EQ−∑k≥1J k[(k − 1)Ψ]1− EP−∑k≥1J k[(k − 1)Ψ]1 . (3.19)Replacing (3.19) in (3.12), we getH(Q | R)−H(P | R) ≥EQ∑k≥1J k[log(q˜kqk)− (k − 1)Ψ]1− EP∑k≥1J k[log(q˜kqk)− (k − 1)Ψ]1 . (3.20)For all k ≥ 1, let ak : R+ × Td → R be given byak(t, x) = log(q˜k(t, x)qk)− (k − 1)Ψ(t, x).From assumptions on q˜k, qk and Ψ, we have aks are deterministic continu-ous functions. Then, it is known from the proof of [2, Theorem 4.25], (inparticular Equation (4.32)) that for all k ∈ N,343.2. Main resultEQ∑k≥1J k[ak]1 = EQ∫ 10∑k≥1⟨q˜k(s)ak(s),Ms−⟩ds ,EP∑k≥1J k[ak]1 = EP∫ 10∑k≥1⟨q˜k(s)ak(s),Ms−⟩ds . (3.21)Now substituting (3.21) in (3.20) we get,H(Q | R)−H(P | R)≥ EQ∫ 10∑k≥1⟨q˜k(s)((k − 1)Ψ(s) + log(q˜k(s)qk)),Ms−⟩ds− EP∫ 10∑k≥1⟨q˜k(s)((k − 1)Ψ(s) + log(q˜k(s)qk)),Ms−⟩ds .(3.22)Given our assumption that h(q˜ | q) < ∞, and Ψ ∈ C2([0, 1] × Td), thereexists a constant C > 0 such that for all s > 0∑k≥1⟨q˜k(s)(log(q˜k(s)qk)− (k − 1)Ψ(s)),Ms−⟩ ≤ C⟨1,Ms−⟩.Now, using the above and (3.14), a similar Fubini argument and definitionof mean measure as in (3.13)  (3.17) in (3.22) we haveH(Q | R)−H(P | R) ≥∫ 10∑k≥1⟨q˜k(s)(log(q˜k(s)qk)− (k − 1)Ψ(s)),EQ [Ms−]⟩ds−∫ 10∑k≥1⟨q˜k(s)(log(q˜k(s)qk)− (k − 1)Ψ(s)),EP [Ms−]⟩ds(3.23)Now, EP [Ms−] = EQ[Ms−] for all s ∈ [0, 1] since the marginals have thesame law under P and Q.353.3. Limitations and Future WorkH(Q | R)−H(P | R) ≥ 0. (3.24)This concludes the result.3.3 Limitations and Future WorkIn Theorem 3.2.1, we assume that the branching mechanism for both P andQ was the same, {q˜k}k∈N. Thus, the entropy minimization method discussedin this chapter can be used to find the law of paths of cells under the as-sumption that the branching mechanism is known. Typically this is not avalid assumption because we only observe snapshots of cells using scRNA-sequencing and we can only estimate the lineage tree using CRISPR. Usingthe lineage tree, it may be possible to find an estimate of the branchingmechanism. Notice that in the proof of the theorem, for (3.23) to hold, it iscrucial that both P and Q have the same branching rate. Our method hasthis limitation which doesn't allow us to handle the general case.In all the literature so far, and also this thesis, we assume that the diffusiv-ity of the Brownian motion is independent of time and the location of thecell. It may be possible that different cell types diffuse differently and thismay cause σ to depend on the location of the cell in gene expression space.The entropy minimization problem with this generalization is an interestingmathematical question.Suppose we have, {µˆti}Ti=1 noisy samples from the ground truth process P .In [13] the authors propose a method, Global Waddington-OT (gWOT),which optimizes in both the marginals Rti and couplings for a regression.This handles the noisy fluctuations due to sampling effects. This providesa method that enables an efficient computational solution for predicting thecell development trajectory. An interesting future direction is to implementthe same data fitting approach for branching Brownian motion with death.36Bibliography[1] Luigi Ambrosio, Luis A Caffarelli, Yann Brenier, Giuseppe Buttazzo,Cedric Villani, Sandro Salsa, Luigi Ambrosio, and Aldo Pratelli. 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Thesis/Dissertation
eng
Mathematics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Angel, Omer
Schiebinger, Geoffrey
Branching Brownian motion models for cell developmental trajectories
Text