David Holcman · Zeev Schuss

Stochastic Narrow Escape in Molecular and Cellular Analysis and Applications Stochastic Narrow Escape in Molecular and Cellular Biology

David Holcman • Zeev Schuss

Stochastic Narrow Escape in Molecular and Cellular Biology Analysis and Applications

123 David Holcman Zeev Schuss Group of Applied Mathematics Department of Mathematics and Tel Aviv University School Ecole Normale Superieure of Mathematical Science Paris, France Tel Aviv, Israel

ISBN 978-1-4939-3102-6 ISBN 978-1-4939-3103-3 (eBook) DOI 10.1007/978-1-4939-3103-3

Library of Congress Control Number: 2015947260

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Preface

This monograph consists of two main parts, mathematical and biological. The math- ematical part, which is quite sophisticated involving advanced asymptotic methods in partial differential equations, is aimed primarily at applied mathematicians and theoretical physicists who are interested in biological applications. The targeted readership of the second part is much wider and includes also computational biologists, theoretical chemists, biochemists, biophysicists, and physiologists. This part does not necessarily require the in-depth digesting of the tough analysis of the first part. It includes a summary of output formulas from the first part and mainly concentrates on their applications in various models of specific problems in theoretical molecular and cellular biology. Stochastic narrow escape consists in the passage of a diffusing particle through a narrow opening in an impermeable wall or the arrival of the trajectory of a diffusion process at a small target. For example, a stochastic narrow escape occurs when an ion diffusing inside a biological cell finds a channel molecule embedded in the cell membrane and squeezes through it across the membrane. A similar situation occurs when a neurotransmitter molecule, such as glutamate, released from a vesicle at the pre-synaptic terminal into a synaptic cleft of a neuron, finds its way by diffusion to a receptor on the post-synaptic terminal of an excitatory synapse and binds to it. Another example is a channel, such as calcium, or a receptor, moving toward its destination in the post-synaptic density (PSD) on a cellular membrane crowded with insurmountable and impermeable obstacles, finds its destination by diffusion through the narrow openings between the obstacles. The mathematical narrow escape problem in stochastic theory is to calculate the mean first passage time (MFPT) of a diffusion process to a small absorbing target (Dirichlet boundary) on the otherwise impermeable (Neumann) boundary of a bounded domain (see Fig. 1.1). The main mathematical effort here is to develop asymptotic methods for the approximate evaluation of the MFPT (also called the narrow escape time, NET) in the various geometries of cellular structures (see Sect. 3.3). The problem is equivalent to the construction of an asymptotic solution to the homogeneous mixed Neumann–Dirichlet boundary value problem for the Poisson equation in a bounded domain in the limit of shrinking Dirichlet part.

vii viii Preface

The NET becomes infinite in this limit, thus rendering its computation a singular perturbation problem. We review in Chaps. 1–3 of this monograph recent developments in the non- standard asymptotics of the problem, which are based on several ingredients: a better resolution of the singularity of Neumann’s function, resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and on the break up of composite domains into simpler compartments. The new methodology applies to two- and higher-dimensional problems. In Chaps. 4–9, we review applications of the narrow escape problem in (see Holcman and Schuss 2013a,b). Critical biological processes, such as synaptic plasticity and transmission, activation of genes by transcription factors, or double-stranded DNA break repair, are controlled by diffusion in structures that have both large and small spatial scales. These may be small binding sites inside or on the surface of the cell, or narrow passages between subcellular compartments. The great disparity in spatial scales is the key to controlling cell function by structure. We report here recent progress on resolving analytical and numerical difficulties in extracting properties from experimental data, from biophysical models, and from Brownian dynamics simulations of diffusion in multi- scale structures. The results of Chaps. 1–3 are applied first to the classification of the various geometries of cellular domains that control ionic and molecular fluxes. The different NETs in various cell geometries are manifested in Brownian dynamics simulations in cellular biology. Specifically, Brownian dynamics simulations can be coarse- grained to the time scale of the NET, thus revealing the dependence of cell function on cell structure. The resolution of the synaptic transmission by solving the NET problem in the synaptic cleft and in the dendritic spine takes advantage of the particular geometries of the synaptic spine and of the synaptic cleft. The explicit asymptotic expression of the NET is applied to stochastic chemical reactions in microdomains, to regulation of calcium flux through the dendritic spine neck, to the delivery of vesicles in neurite outgrowth, to DNA repair in two-dimensional confinement, to control of reactions by hidden binding sites, to asymmetric dumbbell-shaped division in cells, to coarse- graining a stochastic model of a chemical reaction into a Markov chain model, to the coarse-graining of molecular diffusion on a membrane crowded with obstacles to an effective diffusion, as observed in supermicroscopic imaging, to physical virology with a model of the early steps of viral infection in cells, and so on.

Acknowledgements Much of the material presented in this book is based on our collaboration with many scientists and students, whose names are listed next to ours in the author index (A. Singer, J. Reingruber, N. Hoze, K. Dao Duc, N. Rouach, C. Guerrier, T. Lagache, A. Amitai, J. Cartailler, O. Shukroun, J. Yang, A. Taflia, A. Biess, S. Vakeroudis, G. Malherbe, K. Tsaneva, D. Fresche, J. Sibille, I. Kupka, and many others).

Paris, France David Holcman Tel Aviv, Israel Zeev Schuss Contents

1 Elementary Theory of Stochastic Narrow Escape...... 1 1.1 Introduction ...... 1 1.2 The Mixed Boundary Value Problem ...... 2 1.3 Neumann’s Function and an Helmholtz Integral Equation...... 5 1.4 The NET on a Two-Dimensional Riemannian Manifold...... 7 1.4.1 Exit Near Singularities of the Boundary ...... 9 1.4.2 NET on a Two-Dimensional Sphere ...... 11 1.4.3 The Matched Asymptotics Approach ...... 13 1.4.4 Exit Through Several Windows ...... 14 1.4.5 Derivation of the Helmholtz Integral Equation...... 15 1.4.6 Asymptotic Solution of the Helmholtz Equation...... 16 1.4.7 Annotations ...... 20 1.5 NET in Bounded Domains in R3 ...... 21 1.5.1 The Neumann Function in Regular Domains in R3 ...... 21 1.5.2 Elliptic Absorbing Window ...... 22 1.6 Second Order Asymptotics for a Circular Window ...... 25 1.6.1 Higher Order Asymptotics in the Unit Ball ...... 27 1.7 The NET Through Multiple Absorbing Windows ...... 28 1.7.1 Third-Order Asymptotics...... 30 1.8 NET from a General Domain with Two Absorbing Windows ...... 32 1.8.1 Multiple Absorbing Windows ...... 34 1.9 Diffusion Leak in a Conductor ...... 37 1.9.1 Activation Through a Narrow Opening ...... 39 1.9.2 The NET ...... 41 1.9.3 Deep Well: A Markov Chain Model ...... 43 2 Special Asymptotics for Stochastic Narrow Escape ...... 45 2.1 in Dire Straits ...... 47 2.1.1 Exit from Several Bottlenecks ...... 50 2.1.2 Two-Dimensional Bottlenecks ...... 51 2.1.3 Annotations to Sect. 2.1 ...... 54

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2.2 Brownian Needle in Dire Straits ...... 55 2.2.1 The Diffusion Law of a Brownian Needle in a Planar Strip ...... 57 2.2.2 The Turnaround Time ...... 59 2.2.3 The NET in a Solid Funnel-Shaped Domain ...... 64 2.2.4 The MFPT in a Dumbbell-Shaped Domain ...... 69 2.3 The MFPT to a Ribbon ...... 70 2.4 Conditioning and Splitting Probability...... 75 2.4.1 Conditioning ...... 76 2.5 The Splitting Probability for Small Targets ...... 79 2.5.1 The Splitting Probability in an Annulus ...... 79 2.6 Asymptotic Computation of the Dwell Time ...... 82 2.7 NET with Killing...... 85 2.7.1 The Probability of Absorbed Trajectories ...... 85 2.7.2 Decay of the Survival Probability in One Dimension ...... 88 2.7.3 The Ratio of Absorption to Killing ...... 91 2.7.4 Splitting Influx into Absorption and Killing Effluxes ...... 91 3 NET in Molecular and Cellular Biology...... 95 3.1 Introduction ...... 95 3.2 From Molecular to Cellular Description ...... 98 3.2.1 Flux Through Narrow Passages Identifies Cellular Compartments ...... 99 3.2.2 The MFPT and the Survival Probability ...... 100 3.3 Geometrical Classification of Cellular Domains ...... 102 3.3.1 The Formula for the NET N ...... 104 3.3.2 Formulas for Two-Dimensional Domains ...... 105 3.3.3 Formulas for N in Three-Dimensional Domains...... 109 3.4 Annotations...... 111 4 Applications to Cellular Biology and Simulations...... 113 4.1 Applications to Brownian Dynamics Simulations ...... 113 4.2 Diffusion in the Synaptic Cleft ...... 114 4.2.1 Leak of Ions from a Dendritic Spine Neck ...... 116 4.3 Diffusion on a Membrane Crowded with Obstacles...... 116 4.3.1 A Coarse-Grained Model of Membrane Crowding Organization ...... 118 4.3.2 Diffusion of Receptors on the Neuronal Membrane ...... 120 4.4 Synaptic Transmission and the Synaptic Current ...... 121 4.4.1 The Synaptic Cleft ...... 123 4.5 The Mean and Variance of the Synaptic Current Is ...... 125 4.5.1 Properties of the Synaptic Current Is ...... 127 4.5.2 Nonlinearity of the Synaptic Current Is ...... 129 4.6 Coarse-Grained Glutamate Partial Absorption Rate of the PSD...... 130 4.7 Estimate of the Molecular Rate a from Experimental Data ...... 132 4.8 Annotations...... 133 Contents xi

5 Determination of Features from Super-Resolution Data ...... 135 5.1 Receptor Motion on a Cellular Membrane...... 135 5.1.1 The Empirical Moments ...... 136 5.2 Simulations in Empirical Domains ...... 138 5.2.1 Imaging Analysis, Spatial Filtering, and Discretization ...... 138 5.2.2 Residence Time of Receptors in Dendritic Spines ...... 140 5.3 Annotation...... 140 6 Markov Models for Stochastic Chemical Reactions...... 143 6.1 Stochastic Chemical Reactions ...... 143 6.1.1 Introduction to Stochastic Chemical Reactions ...... 143 6.1.2 The Mean Time the Number of Bound Molecules Reaches a Threshold ...... 145 6.2 Applications of MTT in Cell Biology...... 147 6.2.1 Escaping Degradation by Binding microRNAs ...... 147 6.2.2 2D-Markov Chain Model of mRNA Escaping Degradation .. 153 6.3 The Spindle Pole Body Activation ...... 160 6.3.1 Model of the SAC...... 160 6.3.2 Activation of the APC/C ...... 165 6.3.3 Determination of the Production and Unbinding Rates ...... 166 6.4 Annotation...... 168 7 Random Search with Switching ...... 169 7.1 Random Switching Between Two Modes of Diffusion ...... 169 7.1.1 The GNET for a Particle Switching Between Two States .... 171 7.1.2 GNET in One Dimension ...... 172 7.1.3 GNET in Three Dimensions...... 174 7.1.4 Application to the Search by a Transcription Factor ...... 176 7.1.5 Search Analysis for a Three State Switching Process...... 177 7.2 Annotations...... 181 8 Narrow Escape in Other Cellular Processes ...... 183 8.1 Stochastic Model of Vesicular Trafficking During Neurite Outgrowth ...... 183 8.1.1 Derivation of the Join pdfs for a Coarse-Grained Stochastic Equation ...... 186 8.1.2 Calibration Procedure to the Explicit Solution of the FPE ...... 189 8.1.3 Arrival Rate of Vesicles to a Small Opening in the Soma .... 191 8.1.4 Neurite Outgrowth Modulated by Intermittent Neuronal Activity ...... 196 8.2 DNA Repair in a Two-Dimensional Confined Chromatin Structure.. 197 8.3 MFPT to a Hidden Binding Site...... 199 8.4 Asymmetric Dumbbell-Shaped Division in Cells ...... 200 9 Modeling the Early Steps of Viral Infection in Cells ...... 203 9.1 Introduction ...... 203 xii Contents

9.2 Viral Trafficking in the Cell Cytoplasm...... 206 9.2.1 The Steady State Drift for a Two-Dimensional Radial Cell ...... 206 9.2.2 MFPT of a Brownian Motion to the Boundary of a Circular Sector ...... 208 9.2.3 Distribution of Exit Points on @a ...... 209 9.2.4 The Mean Exit Radius ...... 211 9.2.5 Homogenization Procedure to a Constant Drift ...... 212 9.2.6 Asymptotic Solution Versus Brownian Simulations ...... 213 9.2.7 The Cylindrical Geometry...... 214 9.2.8 Conclusion About Homogenization of Stochastic Equations...... 215 9.3 Probability that a Viral Particle Arrives Alive at a Nuclear Pore ..... 216 9.3.1 The Probability PN and Mean Time NN to Reach the Nucleus ...... 216 9.3.2 Probability and MFPT for a Diffusion Process to Reach a Small Nuclear Pore...... 217 9.3.3 Asymptotics of PN for Small Killing Rate k.x/  1 ...... 218 9.3.4 Asymptotics of NN for Small Killing Rate k.x/  1 ...... 222 9.4 The Case of Non-zero Drift ...... 223 9.4.1 Probability to Reach Alive the Nucleus Surface ...... 224 9.4.2 The Arrival Probability PN with the Potential .x/ D Bjxj .. 226 9.4.3 The Conditional MFPT to the Nucleus ...... 228 9.4.4 The Large Degradation Rate Limit k  1...... 230 9.4.5 MFPT of the First DNA Carrier to Reach a Nuclear Pore ...... 234 9.5 Model of the Endosomal Step of Viral Infection ...... 237 9.5.1 Model of the Conformational Change of Hemagglutinin for the Influenza Virus ...... 238 9.6 Annotations...... 244

Bibliography ...... 247

Index ...... 257 List of Figures

Fig. 1.1 Escaping Brownian trajectory ...... 2 Fig. 1.2 Composite domains ...... 3 Fig. 1.3 Receptor movement on the neuronal membrane ...... 4 Fig. 1.4 A small opening near a corner of angle ˛ ...... 10 Fig. 1.5 Escape near a cusp ...... 11 Fig. 1.6 Escape to the north pole ...... 12 Fig. 1.7 Two holes ...... 15 Fig. 1.8 MFPT vs distance...... 19 Fig. 2.1 Domain with a funnel...... 46 Fig. 2.2 Partial block ...... 46 Fig. 2.3 Conformal image of a funnel...... 48 Fig. 2.4 Rod in strip ...... 56 Fig. 2.5 Conformal image of a rod in a strip...... 62 Fig. 2.6 Boundary layers...... 63 Fig. 2.7 Dumbbell-shaped domain ...... 70 Fig. 2.8 Brownian search for a narrow ribbon ...... 72 Fig. 2.9 Conformal mapping ...... 73 Fig. 2.10 Coarse-grained Model of the PSD ...... 80 Fig. 3.1 Spines on a dendrite ...... 97 Fig. 3.2 Electron-microscopy of an excitatory synapse ...... 97 Fig. 3.3 Schematic of a synaptic cleft ...... 98 Fig. 3.4 Schematic of a smooth dendritic spine ...... 98 Fig. 3.5 Brownian motion in a disk ...... 100 Fig. 3.6 Brownian trajectory...... 101 Fig. 3.7 Composite domain ...... 101 Fig. 3.8 Funnel with a finite angle ...... 102 Fig. 3.9 Schematic of a synaptic cleft ...... 104 Fig. 3.10 NET on a surface of revolution ...... 108

xiii xiv List of Figures

Fig. 4.1 Probability that a neurotransmitter reaches a receptor ...... 115 Fig. 4.2 Organization of a neuronal membrane ...... 117 Fig. 4.3 Receptor trajectory on a neuronal membrane ...... 117 Fig. 4.4 Diffusion on a crowded membrane ...... 118 Fig. 4.5 MFPT vs obstacle size...... 119 Fig. 4.6 Organization of the neuronal membrane...... 122 Fig. 4.7 Synapse between neurons ...... 123 Fig. 4.8 Coefficient of variation for the synaptic current...... 127 Fig. 4.9 Optimal PSD radius ...... 128 Fig. 4.10 Simplified model for discrete current levels ...... 130 Fig. 5.1 Schematic representation of the estimation algorithm ...... 137 Fig. 5.2 Spatial filtering...... 139 Fig. 5.3 Residence time of a receptor in a dendritic spine ...... 141 Fig. 6.1 Diffusion with binding sites ...... 146 Fig. 6.2 Schematic miRNA-PTEN post-transcriptional regulation ...... 148 Fig. 6.3 Markov chain diagram ...... 148 Fig. 6.4 Escape probability as a function of the threshold T ...... 153 Fig. 6.5 Markov chain diagram for the mRNA state ...... 155 Fig. 6.6 Markov chain diagram for the mRNA state ...... 156 Fig. 6.7 Escape probability as a function of the threshold ...... 158 Fig. 6.8 Schematic view of the spindle assembly checkpoint...... 161 Fig. 6.9 Markov diagram for the probability of number of Cdc20 molecules...... 162 Fig. 6.10 Range domain for the rate constants ...... 167 Fig. 7.1 Diffusing Brownian ligand in a confined domain ...... 171 Fig. 7.2 Sojourn time for the GNET ...... 173 Fig. 7.3 Exit time for the GNET...... 174 Fig. 7.4 Mean sojourn time for the GNET ...... 176 Fig. 7.5 TF’s motion along the DNA...... 177 Fig. 7.6 Search scenario for a DNA target by a TF...... 178 Fig. 7.7 Search scenario with 3 states ...... 178 Fig. 7.8 Mean search time  ...... 182 Fig. 8.1 Neuron growth ...... 184 Fig. 8.2 Scenario for a vesicle confined to the soma surface...... 185 Fig. 8.3 Steady state distribution ...... 190 Fig. 8.4 Stratification in 2D...... 198 Fig. 8.5 Rod in strip ...... 198 Fig. 8.6 Complex molecule containing a hidden site ...... 199 Fig. 8.7 Dumbbell-shaped dividing cell...... 201 Fig. 9.1 Common entry and uncoating mechanisms of viruses ...... 204 Fig. 9.2 Viral trajectories in cells...... 205 Fig. 9.3 Drift homogenization ...... 207