Statistical biophysics of hematopoiesis and growing cell populations Nathaniel Vincent Mon P`ere Contents 1. Introduction 11 I. Population dynamics of hematopoiesis 19 2. Hematopoiesis: the factory for blood 21 2.1. A brief history of hematopoiesis . 22 2.2. Cellular differentiation . 25 2.2.1. Providing variety and specification . 25 2.2.2. Transitional states and cell fate . 28 2.2.3. The role of cell divisions in differentiation . 30 2.3. Hematopoietic lineages . 31 2.4. Hematopoietic stem cells . 31 2.5. Differentiational tissues accumulate mutations . 33 2.6. Open questions and perspectives . 34 3. Mathematical tools 37 3.1. Stochastic processes . 37 3.1.1. The Bernoulli process . 38 3.1.2. The Poisson process . 40 3.2. Markov Chains . 46 3.2.1. The state space . 47 3.2.2. Markov transition probabilties . 48 3.2.3. Discrete time Markov chains . 49 3 Contents 3.2.4. Continuous time Markov chains . 50 3.2.5. Non-discrete state spaces . 50 3.3. Stochastic population dynamics with Markov chains . 51 3.3.1. The birth-death process . 51 3.3.2. The Moran process . 53 3.4. summary . 55 4. Hematopoietic stem cells: a neutral stochastic population 57 4.1. The importance of stochasticity . 59 4.2. Assumptions for stochastic HSC dynamics . 60 4.2.1. Mutation rate . 64 4.2.2. Division rate . 64 4.3. Modeling the stochastic dynamics of a mutant clone . 65 4.3.1. A birth-death model (is not sufficient) . 65 4.3.2. A Moran model . 67 4.3.3. Moving to real time . 68 4.3.4. The diffusion approximation . 69 5. Evolutionary dynamics of paroxysmal nocturnal hemoglobinuria 73 5.1. Paroxysmal nocturnal hemoglobinuria . 73 5.2. Applying the Moran model . 76 5.2.1. Transition probabilities . 76 5.2.2. Ontogenic growth . 78 5.2.3. Observing multiple clones . 78 5.2.4. Parameter values and diagnosis threshold . 80 5.3. Results and predictions . 82 5.3.1. Probability and prevalence of PNH . 82 5.3.2. Average clone sizes . 85 5.3.3. Arrival times of mutated clone and clinical PNH . 85 5.3.4. Clonal expansion . 86 4 Contents 5.3.5. Disease reduction . 88 5.4. Discussion . 89 5.5. Perspective: HSCs under perturbed hematopoiesis . 91 5.5.1. Feedback driven division rates . 92 5.5.2. Heuristic results . 93 5.6. Conclusion . 96 6. Subclonal dynamics in hematopoietic stem cells 97 6.1. Clonality . 98 6.2. Moran model with asymmetric divisions . 100 6.3. Testing with simulations . 103 6.4. The single cell mutational burden . 104 6.4.1. Mutational burden as a compound Poisson process . 104 6.4.2. Markov chain approach . 106 6.4.3. Discussion: single cell mutatational burden . 108 6.5. The variant allele frequency spectrum (VAF) . 109 6.5.1. Dynamics of the VAF expected value . 110 6.5.2. Dynamics of the VAF variance . 111 6.5.3. Equilibrium distributions . 116 6.5.4. Discussion: VAF . 116 6.6. The sampling problem . 118 6.7. Applications to a human HSC dataset . 121 6.7.1. Data: somatic mutations in single HSCs . 121 6.7.2. Single cell mutational burden . 122 6.7.3. Variant allele frequency spectrum: fitting parameters with Ap- proximate Bayesion Computation . 122 6.7.4. Discussion: applications to a dataset . 127 6.8. Conclusions and perspective . 127 5 Contents 7. Feedback-driven compartmental dynamics of hematopoiesis 129 7.1. A compartmental model of hematopoiesis . 132 7.1.1. Dingli model . 132 7.1.2. Introducing feedback . 134 7.2. Analysis . 136 7.2.1. Sequential coupling elicits three types of behavior . 136 7.2.2. Increasing cell amplification between compartments reduces stability138 7.2.3. Recovery time as a measure of efficiency . 140 7.2.4. Inclusion of feedback allows prediction of erythrocyte dynamics . 140 7.2.5. Chronic perturbations lead to new equilibrium states . 143 7.3. Discussion and conclusions . 144 II. Statistical mechanics of proliferating cells 167 8. Cell movement as a stochastic process 169 8.1. Motility in cancer: a motivating example . 171 8.2. Cells as motile particles . 175 8.3. Basics of stochastic motion . 176 8.3.1. Brownian motion . 177 8.3.2. Generalizations and other models . 183 9. Stochastic motion under population growth 185 9.1. The problem of growth . 185 9.2. Brownian motion in an ideal gas . 186 9.2.1. Velocity correlation of the random walk . 187 9.3. Coupling the Brownian Langevin equation to the particle density . 189 9.3.1. Fixed density populations . 189 9.3.2. Growing populations . 190 9.3.3. Alternative types of motion . 191 6 Contents 9.4. Comparison of the Langevin equation with direct particle simulations . 192 9.4.1. Fixed density results . 193 9.4.2. Growing population results . 198 9.5. Perspective: localizing the LE for interacting particles . 198 9.6. Discussion . 200 10.Conclusions 213 10.1. Population dynamics of hematopoiesis . 213 10.2. Statistical mechanics of proliferating cells . 217 A. Population dynamics of hematopoiesis 221 A.1. Combining Poisson processes . 221 A.2. Simulations of the Moran model with mutant accumulation . 222 A.2.1. The cell population . 223 A.2.2. Events which alter the population . 223 A.2.3. Mutations . 224 A.2.4. Time evolution . 224 A.3. Obtaining the mean and variance of the compound Poisson distribution . 224 A.4. Compartment model of hematopoiesis: fixing parameter values . 225 B. Statistical mechanics of cell motion 227 B.1. Particle simulation . 227 B.1.1. Particle properties . 227 B.1.2. Particle collisions . 228 B.1.3. Confined space: minimum image periodic boundaries . 228 B.1.4. Accounting for center of mass drift . 228 B.1.5. Population growth . 229 B.1.6. Sketch of the simulation algorithm . 229 B.2. Numerically simulating the Langevin equation . 230 7.