1. Review: Cancer Modeling
Natalia Komarova
(University of California - Irvine)

2. Plan Introduction: The concept of somatic evolution
Loss-of-function and gain-of-function mutations
Mass-action modeling
Spatial modeling
Hierarchical modeling
Consequences from the point of view of tissue architecture and homeostatic control

3. Darwinian evolution (of species)
Time-scale: hundreds of millions of years
Organisms reproduce and die in an environment with shared resources

4. Darwinian evolution (of species)
Time-scale: hundreds of millions of years
Organisms reproduce and die in an environment with shared resources
Inheritable germline mutations (variability)
Selection
(survival of the fittest)

5. Somatic evolution
Cells reproduce and die inside an organ of one organism
Time-scale: tens of years

6. Somatic evolution
Cells reproduce and die inside an organ of one organism
Time-scale: tens of years
Inheritable mutations in cells’ genomes (variability)
Selection
(survival of the fittest)

7. Cancer as somatic evolution
Cells in a multicellular organism have evolved to co-operate and perform their respective functions for the good of the whole organism

8. Cancer as somatic evolution
Cells in a multicellular organism have evolved to co-operate and perform their respective functions for the good of the whole organism
A mutant cell that “refuses” to co-operate may have a selective advantage

9. Cancer as somatic evolution
Cells in a multicellular organism have evolved to co-operate and perform their respective functions for the good of the whole organism
A mutant cell that “refuses” to co-operate may have a selective advantage
The offspring of such a cell may spread

10. Cancer as somatic evolution
Cells in a multicellular organism have evolved to co-operate and perform their respective functions for the good of the whole organism
A mutant cell that “refuses” to co-operate may have a selective advantage
The offspring of such a cell may spread
This is a beginning of cancer

11. Progression to cancer

12. Progression to cancer
Constant population

13. Progression to cancer
Advantageous mutant

14. Progression to cancer
Clonal expansion

15. Progression to cancer
Saturation

16. Progression to cancer
Advantageous mutant

17. Progression to cancer
Wave of clonal expansion

18. Genetic pathways to colon cancer (Bert Vogelstein)
“Multi-stage carcinogenesis”

19. Methodology: modeling a colony of cells
Cells can divide, mutate and die

20. Methodology: modeling a colony of cells
Cells can divide, mutate and die
Mutations happen according to a “mutation-selection diagram”, e.g. u1 u2 u3 u4
(1)
(r1)
(r2)
(r3)
(r4)

21. Mutation-selection network
(1)
(r1)
(r4)
(r6) u2 u2 u5 u8
(r1)
(r5)
(r7)

22. Common patterns in cancer progression
Gain-of-function mutations
Loss-of-function mutations

23. Gain-of-function mutations
Oncogenes
K-Ras (colon cancer), Bcr-Abl (CML leukemia)
Increased fitness of the resulting type
Wild type
Oncogene u
(1)
(r)

24. Loss-of-function mutations
Tumor suppressor genes
APC (colon cancer), Rb (retinoblastoma), p53 (many cancers)
Neutral or disadvantageous intermediate mutants
Increased fitness of the resulting type
Wild type
TSP+/+
TSP+/-
TSP-/- u u x x x
(1)
(r<1)
(R>1)

25. Stochastic dynamics on a selection-mutation network
Given a selection-mutation diagram
Assume a constant population with a cellular turn-over
Define a stochastic birth-death process with mutations
Calculate the probability and timing of mutant generation

26. Gain-of-function mutations
Selection-mutation diagram:
Number of is i u
(1)
(r )
Number of is j=N-i
Fitness = 1
Fitness = r >1

27. Evolutionary selection dynamics
Fitness = 1
Fitness = r >1

28. Evolutionary selection dynamics
Fitness = 1
Fitness = r >1

29. Evolutionary selection dynamics
Fitness = 1
Fitness = r >1

30. Evolutionary selection dynamics
Fitness = 1
Fitness = r >1

31. Evolutionary selection dynamics
Fitness = 1
Fitness = r >1

32. Evolutionary selection dynamics
Start from only one cell of the second type;
Suppress further mutations.
What is the chance that it will take over?
Fitness = 1
Fitness = r >1

33. Evolutionary selection dynamics
Start from only one cell of the second type.
What is the chance that it will take over?
If r=1 then = 1/N
If r<1 then < 1/N
If r>1 then > 1/N
If r then = 1
Fitness = 1
Fitness = r >1

34. Evolutionary selection dynamics
Start from zero cell of the second type.
What is the expected time until the second type
takes over?
Fitness = 1
Fitness = r >1

35. Evolutionary selection dynamics
Start from zero cell of the second type.
What is the expected time until the second type
takes over?
In the case of rare mutations,
we can show that
Fitness = 1
Fitness = r >1

36. Loss-of-function mutations
(1)
(r)
(a)
What is the probability that by time t a mutant of
has been created?
Assume that and

37. 1D Markov process j is the random variable,
If j = 1,2,…,N then there are j intermediate mutants, and no double-mutants
If j=E, then there is at least one double-mutant
j=E is an absorbing state

38. Transition probabilities

39. A two-step process

40. A two-step process

41. A two step process … …

42. A two-step process (1) (r) (a) Scenario 1:
gets fixated first, and then a mutant of
is created;
Number of cells
time

43. Stochastic tunneling …

44. Stochastic tunneling (1) (r) (a) Scenario 2:
A mutant of is created before reaches fixation
Number of cells
time

45. The coarse-grained description
Long-lived states:
x0 …“all green”
x1 …“all blue”
x2 …“at least one red”

46. Stochastic tunneling Assume that and Neutral intermediate mutant
Disadvantageous intermediate mutant
Assume that and

47. The mass-action model is unrealistic
All cells are assumed to interact with each other, regardless of their spatial location
All cells of the same type are identical

48. The mass-action model is unrealistic
All cells are assumed to interact with each other, regardless of their spatial location
Spatial model of cancer
All cells of the same type are identical

49. The mass-action model is unrealistic
All cells are assumed to interact with each other, regardless of their spatial location
Spatial model of cancer
All cells of the same type are identical
Hierarchical model of cancer

50. Spatial model of cancer
Cells are situated in the nodes of a regular, one-dimensional grid
A cell is chosen randomly for death
It can be replaced by offspring of its two nearest neighbors

51. Spatial dynamics

52. Spatial dynamics

53. Spatial dynamics

54. Spatial dynamics

55. Spatial dynamics

56. Spatial dynamics

57. Spatial dynamics

58. Spatial dynamics

59. Spatial dynamics

60. Gain-of-function: probability to invade
In the spatial model, the probability to invade depends on the spatial location of the initial mutation

61. Probability of invasion
Neutral
mutants, r = 1
Advantageous
mutants, r = 1.2
Mass-action
Disadvantageous
mutants, r = 0.95
Spatial

62. Use the periodic boundary conditions
Mutant island

63. Probability to invade For disadvantageous mutants For neutral mutants
For advantageous mutants

64. Loss-of-function mutations
(1)
(r)
(a)
What is the probability that by time t a mutant of
has been created?
Assume that and

65. Transition probabilities
No double-mutants,
j intermediate cells
At least one double-mutant
Mass-action
Space

66. Stochastic tunneling

67. Stochastic tunneling
Slower

68. Stochastic tunneling
Slower
Faster

69. The mass-action model is unrealistic
All cells are assumed to interact with each other, regardless of their spatial location
Spatial model of cancer
All cells of the same type are identical
Hierarchical model of cancer P

70. Hierarchical model of cancer

71. Colon tissue architecture

72. Colon tissue architecture
Crypts of a colon

73. Colon tissue architecture
Crypts of a colon

74. Cancer of epithelial tissues
Gut
Cells in a crypt of a colon

75. Cancer of epithelial tissues
Cells in a crypt of a colon
Gut
Stem cells replenish the
tissue; asymmetric divisions

76. Cancer of epithelial tissues
Cells in a crypt of a colon
Gut
Proliferating cells divide
symmetrically and
differentiate
Stem cells replenish the
tissue; asymmetric divisions

77. Cancer of epithelial tissues
Cells in a crypt of a colon
Gut
Differentiated cells get
shed off into the lumen
Proliferating cells divide
symmetrically and
differentiate
Stem cells replenish the
tissue; asymmetric divisions

78. Finite branching process

79. Cellular origins of cancer
Gut
If a stem cell tem cell
acquires a mutation,
the whole crypt is
transformed

80. Cellular origins of cancer
Gut
If a daughter cell acquires
a mutation, it will probably
get washed out before
a second mutation can hit

81. Colon cancer initiation

82. Colon cancer initiation

83. Colon cancer initiation

84. Colon cancer initiation

85. Colon cancer initiation

86. Colon cancer initiation

87. First mutation in a daughter cell

88. First mutation in a daughter cell

89. First mutation in a daughter cell

90. First mutation in a daughter cell

91. First mutation in a daughter cell

92. First mutation in a daughter cell

93. First mutation in a daughter cell

94. First mutation in a daughter cell

95. First mutation in a daughter cell

96. First mutation in a daughter cell

97. First mutation in a daughter cell

98. First mutation in a daughter cell

99. Two-step process and tunneling
time
Number of cells
First hit in the stem cell
Second hit in a
daughter cell
Number of cells
First hit in a daughter cell
time

100. Stochastic tunneling in a hierarchical model

101. Stochastic tunneling in a hierarchical model
The same

102. Stochastic tunneling in a hierarchical model
The same
Slower

103. The mass-action model is unrealistic
All cells are assumed to interact with each other, regardless of their spatial location
Spatial model of cancer
All cells of the same type are identical
Hierarchical model of cancer P P

104. Comparison of the models
Probability of mutant invasion for
gain-of-function mutations
r = 1 neutral

105. Comparison of the models
The tunneling rate
(lowest rate)

106. The tunneling and two-step regimes

107. Production of double-mutants
Population size
Small
Large
Interm. mutants
Neutral
(mass-action,
spatial and
hierarchical)
All models give
the same results
Spatial model is the fastest
Hierarchical model is the
slowest
Disadvantageous
(mass-action and
Spatial only)
Spatial model is the
fastest
Mass-action model is
faster
Spatial model is
slower

108. Production of double-mutants
Population size
Small
Large
Interm. mutants
Neutral
(mass-action,
spatial and
hierarchical)
All models give
the same results
Spatial model is the fastest
Hierarchical model is the
slowest
Disadvantageous
(mass-action and
Spatial only)
Spatial model is the
fastest
Mass-action model is
faster
Spatial model is
slower

109. The definition of “small”
1000
r=1
Spatial model is the fastest
r=0.99
500
r=0.95
r=0.8

110. Summary
The details of population modeling are important, the simple mass-action can give wrong predictions

111. Summary
The details of population modeling are important, the simple mass-action can give wrong predictions
Different types of homeostatic control have different consequence in the context of cancerous transformation

112. Summary
If the tissue is organized into compartments with stem cells and daughter cells, the risk of mutations is lower than in homogeneous populations

113. Summary
If the tissue is organized into compartments with stem cells and daughter cells, the risk of mutations is lower than in homogeneous populations
For population sizes greater than 102 cells, spatial “nearest neighbor” model yields the lowest degree of protection against cancer

114. Summary
A more flexible homeostatic regulation mechanism with an increased cellular motility will serve as a protection against double-mutant generation

115. Conclusions
Main concept: cancer is a highly structured evolutionary process
Main tool: stochastic processes on selection-mutation networks
We studied the dynamics of gain-of-function and loss-of-function mutations
There are many more questions in cancer research…