American Institute of Mathematical Sciences

Advanced
2013, 10(3): 761-775. doi: 10.3934/mbe.2013.10.761

Spatial stochastic models of cancer: Fitness, migration, invasion

Natalia L. Komarova 1,

1. 

Department of Mathematics, University of California Irvine, Irvine CA 92697, United States

Received  July 2012 Revised  November 2012 Published  April 2013

Cancer progression is driven by genetic and epigenetic events giving rise to heterogeneity of cell phenotypes, and by selection forces that shape the changing composition of tumors. The selection forces are dynamic and depend on many factors. The cells favored by selection are said to be more ``fit'' than others. They tend to leave more viable offspring and spread through the population. What cellular characteristics make certain cells more fit than others? What combinations of the mutant characteristics and ``background'' characteristics make the mutant cells win the evolutionary competition? In this review we concentrate on two phenotypic characteristics of cells: their reproductive potential and their motility. We show that migration has a direct positive impact on the ability of a single mutant cell to invade a pre-existing colony. Thus, a decrease in the reproductive potential can be compensated by an increase in cell migration. We further demonstrate that the neutral ridges (the set of all types with the invasion probability equal to that of the host cells) remain invariant under the increase of system size (for large system sizes), thus making the invasion probability a universal characteristic of the cells' selection status. We list very general conditions under which the optimal phenotype is just one single strategy (thus leading to a nearly-homogeneous type invading the colony), or a large set of strategies that differ by their reproductive potentials and migration characteristics, but have a nearly-equal fitness. In the latter case the evolutionary competition will result in a highly heterogeneous population.
Keywords: spatial cell dynamics, fitness, agent-based modeling, lattice models, Cellular automata, cell motolity..
Mathematics Subject Classification: 92B05, 92C17, 92C50, 92D2.
Citation: Natalia L. Komarova. Spatial stochastic models of cancer: Fitness, migration, invasion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 761-775. doi: 10.3934/mbe.2013.10.761
References:
[1]

A. Anderson, M. Chaplain, K. Rejniak and J. Fozard, Single-cell based models in biology and medicine,, Math. Med. Biol., 25 (2008), 185.   Google Scholar

[2]

A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nature Reviews Cancer, 8 (2008), 227.   Google Scholar

[3]

R. M. Anderson and R. M. May, Coevolution of hosts and parasites,, Parasitology, 85 (1982), 411.   Google Scholar

[4]

M. Boots, P. J. Hudson and A. Sasaki, Large shifts in pathogen virulence relate to host population structure,, Science, 303 (2004), 842.   Google Scholar

[5]

J. Breivik and G. Gaudernack, Carcinogenesis and natural selection: A new perspective to the genetics and epigenetics of colorectal cancer,, Adv. Cancer Res., 76 (1999), 187.   Google Scholar

[6]

H. Byrne, T. Alarcón, M. Owen, S. Webb and P. Maini, Modeling aspects of cancer dynamics: A review,, Phi. Trans. R. Soc. A, 364 (2006), 1563.  doi: 10.1098/rsta.2006.1786.  Google Scholar

[7]

A. Chauvière, L. Preziosi and C. Verdier, "Cell Mechanics: From Single Scale-Based Models to Multiscale Modeling,", CRC Press, 32 (2009).  doi: 10.1201/9781420094558.  Google Scholar

[8]

B. Chopard, R. Ouared, A. Deutsch, H. Hatzikirou and D. Wolf-Gladrow, Lattice-gas cellular automaton models for biology: from fluids to cells,, Acta Biotheoretica, 58 (2010), 329.   Google Scholar

[9]

T. Deisboeck and G. Stamatakos, "Multiscale Cancer Modeling,", CRC Press, 34 (2010).   Google Scholar

[10]

T. Deisboeck, L. Zhang, J. Yoon and J. Costa, In silico cancer modeling: is it ready for prime time?,, in press., ().   Google Scholar

[11]

A. Deutsch and S. Dormann, "Cellular Automaton Modeling of Biological Pattern Formation,", Birkhauser, (2005).   Google Scholar

[12]

D. Drasdo and S. Höhme, On the role of physics in the growth and pattern of multicellular systems: What we learn from individual-cell based models?,, J. Stat. Phys., 128 (2007), 287.  doi: 10.1007/s10955-007-9289-x.  Google Scholar

[13]

D. Ebert and E. A. Herre, The evolution of parasitic diseases,, Parasitol Today, 12 (1996), 96.   Google Scholar

[14]

D. Ebert and K. L. Mangin, The influence of host demography on the evolution of virulence of a microsporidian gut parasite,, Evolution, 51 (1997), 1828.   Google Scholar

[15]

A. Fasano, A. Bertuzzi and A. Gandolfi, Complex systems in biomedicine chapter mathematical modelling of tumour growth and treatment,, Milan: Springer, (2006), 71.  doi: 10.1007/88-470-0396-2_3.  Google Scholar

[16]

S. A. Frank, Models of parasite virulence,, Q. Rev. Biol., 71 (1996), 37.   Google Scholar

[17]

J. Galle, G. Aust, G. Schaller, T. Beyer and D. Drasdo, Individual cell-based models of the spatial temporal organization of multicellular systems- achievements and limitations,, Cytometry, 69A (2006), 704.   Google Scholar

[18]

R. Gatenby and P. Maini, Mathematical oncology: Cancer summed up,, Nature, 421 (2003).   Google Scholar

[19]

D. Hanahan and R. Weinberg, The hallmarks of cancer,, CELL, 100 (2000), 57.   Google Scholar

[20]

P. Hinow,, P. Gerlee, L. McCawley, V. Quaranta, M. Ciobanu, S. Wang,, J. Graham, B. Ayati, J. Claridge, K. Swanson, et al., A spatial model of tumor-host interaction: Application of chemotherapy,, Mathematical Biosciences and Engineering: MBE, 6 (2009).  doi: 10.3934/mbe.2009.6.521.  Google Scholar

[21]

Y. Iwasa, F. Michor and M. A. Nowak, Stochastic tunnels in evolutionary dynamics,, Genetics, 166 (2004), 1571.   Google Scholar

[22]

Y. Jiao and S. Torquato, A cellular automaton model for tumor growth in heterogeneous environment,, Bulletin of the American Physical Society, 56 (2011).   Google Scholar

[23]

N. Komarova, Loss- and gain-of-function mutations in cancer: Mass-action, spatial and hierarchical models,, Jour. Stat. Phys., 128 (2007), 413.  doi: 10.1007/s10955-006-9238-0.  Google Scholar

[24]

N. L. Komarova, Spatial stochastic models for cancer initiation and progression,, Bull. Math. Biol., 68 (2006), 1573.  doi: 10.1007/s11538-005-9046-8.  Google Scholar

[25]

N. L. Komarova, A. Sengupta and M. A. Nowak, Mutation-selection networks of cancer initiation: Tumor suppressor genes and chromosomal instability,, J. Theor. Biol., 223 (2003), 433.  doi: 10.1016/S0022-5193(03)00120-6.  Google Scholar

[26]

B. R. Levin, The evolution and maintenance of virulence in microparasites,, Emerg. Infect. Dis., 2 (1996), 93.   Google Scholar

[27]

L. Merlo, J. Pepper, B. Reid and C. Maley, Cancer as an evolutionary and ecological process,, Nat. Rev. Cancer, 6 (2006), 924.   Google Scholar

[28]

F. Michor, Y. Iwasa, H. Rajagopalan, C. Lengauer and M. A. Nowak, Linear model of colon cancer initiation,, Cell Cycle, 3 (2004), 358.   Google Scholar

[29]

P. Moran, "The Statistical Processes of Evolutionary Theory,", Clarendon, (1962).   Google Scholar

[30]

M. Nowak and K. Sigmund, Evolutionary dynamics of biological games,, Science, 303 (2004), 793.   Google Scholar

[31]

M. A. Nowak, N. L. Komarova, A. Sengupta, P. V. Jallepalli, I.-M. Shih, B. Vogelstein and C. Lengauer, The role of chromosomal instability in tumor initiation,, Proc. Natl. Acad. Sci. U S A, 99 (2002), 16226.   Google Scholar

[32]

M. A. Nowak and R. M. May, Superinfection and the evolution of parasite virulence,, Proc. Biol. Sci., 255 (1994), 81.   Google Scholar

[33]

M. A. Nowak, F. Michor, N. L. Komarova and Y. Iwasa, Evolutionary dynamics of tumor suppressor gene inactivation,, Proc. Natl. Acad. Sci. U S A, 101 (2004), 10635.   Google Scholar

[34]

P. Nowell, The clonal evolution of tumor cell populations,, Science, 194 (1976), 23.   Google Scholar

[35]

V. Quaranta, K. Rejniak, P. Gerlee and A. Anderson, Invasion emerges from cancer cell adaptation to competitive microenvironments: Quantitative predictions from multiscale mathematical models,, Sem. Cancer Biol., (2008).   Google Scholar

[36]

K. Rejniak and A. Anderson, Hybrid models of tumor growth,, Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 3 (2011), 115.   Google Scholar

[37]

C. W. Rinker-Schaeffer, J. P. O'Keefe, D. R. Welch and D. Theodorescu, Metastasis suppressor proteins: Discovery, molecular mechanisms, and clinical application,, CLINICAL CANCER RESEARCH, 12 (2006), 3882.   Google Scholar

[38]

J. Sagotsky and T. Deisboeck, Simulating cancer growth with agent-based models,, Multiscale Cancer Modeling, 34 (2010).   Google Scholar

[39]

A. Sasaki and M. Boots, Parasite evolution and extinctions,, Ecology Letters, 6 (2003).   Google Scholar

[40]

J. M. Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982).   Google Scholar

[41]

C. Thalhauser, J. Lowengrub, D. Stupack and N. Komarova, Research selection in spatial stochastic models of cancer: Migration as a key modulator of fitness,, Biology Direct, 5 (2010).   Google Scholar

[42]

T. L. Vincent and J. S. Brown, "Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics,", Cambridge University Press, (2005).   Google Scholar

[43]

P. Vineis and M. Berwick, The population dynamics of cancer: A Darwinian perspective,, Int. J. Epidemiol, 35 (2006), 1151.   Google Scholar

[44]

D. Wodarz and N. Komarova, "Computational Biology of Cancer: Lecture Notes and Mathematical Modeling,", World Scientific, (2005).   Google Scholar

[45]

S. Wright, The roles of mutation, inbreeding, crossbreeding, and selection in evolution,, in, (1932), 355.   Google Scholar

[46]

L. Zhang, Z. Wang, J. Sagotsky and T. Deisboeck, Multiscale agent-based cancer modeling,, Journal of Mathematical Biology, 58 (2009), 545.  doi: 10.1007/s00285-008-0211-1.  Google Scholar

show all references

References:
[1]

A. Anderson, M. Chaplain, K. Rejniak and J. Fozard, Single-cell based models in biology and medicine,, Math. Med. Biol., 25 (2008), 185.   Google Scholar

[2]

A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nature Reviews Cancer, 8 (2008), 227.   Google Scholar

[3]

R. M. Anderson and R. M. May, Coevolution of hosts and parasites,, Parasitology, 85 (1982), 411.   Google Scholar

[4]

M. Boots, P. J. Hudson and A. Sasaki, Large shifts in pathogen virulence relate to host population structure,, Science, 303 (2004), 842.   Google Scholar

[5]

J. Breivik and G. Gaudernack, Carcinogenesis and natural selection: A new perspective to the genetics and epigenetics of colorectal cancer,, Adv. Cancer Res., 76 (1999), 187.   Google Scholar

[6]

H. Byrne, T. Alarcón, M. Owen, S. Webb and P. Maini, Modeling aspects of cancer dynamics: A review,, Phi. Trans. R. Soc. A, 364 (2006), 1563.  doi: 10.1098/rsta.2006.1786.  Google Scholar

[7]

A. Chauvière, L. Preziosi and C. Verdier, "Cell Mechanics: From Single Scale-Based Models to Multiscale Modeling,", CRC Press, 32 (2009).  doi: 10.1201/9781420094558.  Google Scholar

[8]

B. Chopard, R. Ouared, A. Deutsch, H. Hatzikirou and D. Wolf-Gladrow, Lattice-gas cellular automaton models for biology: from fluids to cells,, Acta Biotheoretica, 58 (2010), 329.   Google Scholar

[9]

T. Deisboeck and G. Stamatakos, "Multiscale Cancer Modeling,", CRC Press, 34 (2010).   Google Scholar

[10]

T. Deisboeck, L. Zhang, J. Yoon and J. Costa, In silico cancer modeling: is it ready for prime time?,, in press., ().   Google Scholar

[11]

A. Deutsch and S. Dormann, "Cellular Automaton Modeling of Biological Pattern Formation,", Birkhauser, (2005).   Google Scholar

[12]

D. Drasdo and S. Höhme, On the role of physics in the growth and pattern of multicellular systems: What we learn from individual-cell based models?,, J. Stat. Phys., 128 (2007), 287.  doi: 10.1007/s10955-007-9289-x.  Google Scholar

[13]

D. Ebert and E. A. Herre, The evolution of parasitic diseases,, Parasitol Today, 12 (1996), 96.   Google Scholar

[14]

D. Ebert and K. L. Mangin, The influence of host demography on the evolution of virulence of a microsporidian gut parasite,, Evolution, 51 (1997), 1828.   Google Scholar

[15]

A. Fasano, A. Bertuzzi and A. Gandolfi, Complex systems in biomedicine chapter mathematical modelling of tumour growth and treatment,, Milan: Springer, (2006), 71.  doi: 10.1007/88-470-0396-2_3.  Google Scholar

[16]

S. A. Frank, Models of parasite virulence,, Q. Rev. Biol., 71 (1996), 37.   Google Scholar

[17]

J. Galle, G. Aust, G. Schaller, T. Beyer and D. Drasdo, Individual cell-based models of the spatial temporal organization of multicellular systems- achievements and limitations,, Cytometry, 69A (2006), 704.   Google Scholar

[18]

R. Gatenby and P. Maini, Mathematical oncology: Cancer summed up,, Nature, 421 (2003).   Google Scholar

[19]

D. Hanahan and R. Weinberg, The hallmarks of cancer,, CELL, 100 (2000), 57.   Google Scholar

[20]

P. Hinow,, P. Gerlee, L. McCawley, V. Quaranta, M. Ciobanu, S. Wang,, J. Graham, B. Ayati, J. Claridge, K. Swanson, et al., A spatial model of tumor-host interaction: Application of chemotherapy,, Mathematical Biosciences and Engineering: MBE, 6 (2009).  doi: 10.3934/mbe.2009.6.521.  Google Scholar

[21]

Y. Iwasa, F. Michor and M. A. Nowak, Stochastic tunnels in evolutionary dynamics,, Genetics, 166 (2004), 1571.   Google Scholar

[22]

Y. Jiao and S. Torquato, A cellular automaton model for tumor growth in heterogeneous environment,, Bulletin of the American Physical Society, 56 (2011).   Google Scholar

[23]

N. Komarova, Loss- and gain-of-function mutations in cancer: Mass-action, spatial and hierarchical models,, Jour. Stat. Phys., 128 (2007), 413.  doi: 10.1007/s10955-006-9238-0.  Google Scholar

[24]

N. L. Komarova, Spatial stochastic models for cancer initiation and progression,, Bull. Math. Biol., 68 (2006), 1573.  doi: 10.1007/s11538-005-9046-8.  Google Scholar

[25]

N. L. Komarova, A. Sengupta and M. A. Nowak, Mutation-selection networks of cancer initiation: Tumor suppressor genes and chromosomal instability,, J. Theor. Biol., 223 (2003), 433.  doi: 10.1016/S0022-5193(03)00120-6.  Google Scholar

[26]

B. R. Levin, The evolution and maintenance of virulence in microparasites,, Emerg. Infect. Dis., 2 (1996), 93.   Google Scholar

[27]

L. Merlo, J. Pepper, B. Reid and C. Maley, Cancer as an evolutionary and ecological process,, Nat. Rev. Cancer, 6 (2006), 924.   Google Scholar

[28]

F. Michor, Y. Iwasa, H. Rajagopalan, C. Lengauer and M. A. Nowak, Linear model of colon cancer initiation,, Cell Cycle, 3 (2004), 358.   Google Scholar

[29]

P. Moran, "The Statistical Processes of Evolutionary Theory,", Clarendon, (1962).   Google Scholar

[30]

M. Nowak and K. Sigmund, Evolutionary dynamics of biological games,, Science, 303 (2004), 793.   Google Scholar

[31]

M. A. Nowak, N. L. Komarova, A. Sengupta, P. V. Jallepalli, I.-M. Shih, B. Vogelstein and C. Lengauer, The role of chromosomal instability in tumor initiation,, Proc. Natl. Acad. Sci. U S A, 99 (2002), 16226.   Google Scholar

[32]

M. A. Nowak and R. M. May, Superinfection and the evolution of parasite virulence,, Proc. Biol. Sci., 255 (1994), 81.   Google Scholar

[33]

M. A. Nowak, F. Michor, N. L. Komarova and Y. Iwasa, Evolutionary dynamics of tumor suppressor gene inactivation,, Proc. Natl. Acad. Sci. U S A, 101 (2004), 10635.   Google Scholar

[34]

P. Nowell, The clonal evolution of tumor cell populations,, Science, 194 (1976), 23.   Google Scholar

[35]

V. Quaranta, K. Rejniak, P. Gerlee and A. Anderson, Invasion emerges from cancer cell adaptation to competitive microenvironments: Quantitative predictions from multiscale mathematical models,, Sem. Cancer Biol., (2008).   Google Scholar

[36]

K. Rejniak and A. Anderson, Hybrid models of tumor growth,, Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 3 (2011), 115.   Google Scholar

[37]

C. W. Rinker-Schaeffer, J. P. O'Keefe, D. R. Welch and D. Theodorescu, Metastasis suppressor proteins: Discovery, molecular mechanisms, and clinical application,, CLINICAL CANCER RESEARCH, 12 (2006), 3882.   Google Scholar

[38]

J. Sagotsky and T. Deisboeck, Simulating cancer growth with agent-based models,, Multiscale Cancer Modeling, 34 (2010).   Google Scholar

[39]

A. Sasaki and M. Boots, Parasite evolution and extinctions,, Ecology Letters, 6 (2003).   Google Scholar

[40]

J. M. Smith, "Evolution and the Theory of Games,", Cambridge University Press, (1982).   Google Scholar

[41]

C. Thalhauser, J. Lowengrub, D. Stupack and N. Komarova, Research selection in spatial stochastic models of cancer: Migration as a key modulator of fitness,, Biology Direct, 5 (2010).   Google Scholar

[42]

T. L. Vincent and J. S. Brown, "Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics,", Cambridge University Press, (2005).   Google Scholar

[43]

P. Vineis and M. Berwick, The population dynamics of cancer: A Darwinian perspective,, Int. J. Epidemiol, 35 (2006), 1151.   Google Scholar

[44]

D. Wodarz and N. Komarova, "Computational Biology of Cancer: Lecture Notes and Mathematical Modeling,", World Scientific, (2005).   Google Scholar

[45]

S. Wright, The roles of mutation, inbreeding, crossbreeding, and selection in evolution,, in, (1932), 355.   Google Scholar

[46]

L. Zhang, Z. Wang, J. Sagotsky and T. Deisboeck, Multiscale agent-based cancer modeling,, Journal of Mathematical Biology, 58 (2009), 545.  doi: 10.1007/s00285-008-0211-1.  Google Scholar

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