Statistical biophysics of hematopoiesis and
growing cell populations
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
3.2.4. 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
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
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
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
