# From invasion to latency: intracellular noise and cell motility as key controls of the competition between resource-limited cellular populations

- 322 Downloads
- 8 Citations

## Abstract

In this paper we analyse stochastic models of the competition between two resource-limited cell populations which differ in their response to nutrient availability: the resident population exhibits a switch-like response behaviour while the invading population exhibits a bistable response. We investigate how noise in the intracellular regulatory pathways and cell motility influence the fate of the incumbent and invading populations. We focus initially on a spatially homogeneous system and study in detail the role of intracellular noise. We show that in such well-mixed systems, two distinct regimes exist: In the low (intracellular) noise limit, the invader has the ability to invade the resident population, whereas in the high noise regime competition between the two populations is found to be neutral and, in accordance with neutral evolution theory, invasion is a random event. Careful examination of the system dynamics leads us to conclude that (i) even if the invader is unable to invade, the distribution of survival times, \(P_S(t)\), has a fat-tail behaviour (\(P_S(t)\sim t^{-1}\)) which implies that small colonies of mutants can coexist with the resident population for arbitrarily long times, and (ii) the bistable structure of the invading population increases the stability of the latent population, thus increasing their long-term likelihood of survival, by decreasing the intensity of the noise at the population level. We also examine the effects of spatial inhomogeneity. In the low noise limit we find that cell motility is positively correlated with the aggressiveness of the invader as defined by the time the invader takes to invade the resident population: the faster the invasion, the more aggressive the invader.

## Keywords

Invasion Latency Noise Motility## Mathematics Subject Classification

92D25 97M60 60J80## Notes

### Acknowledgments

PG and TA gratefully acknowledge the Spanish Ministry for Science and Innovation (MICINN) for funding under grants MTM2008-05271, MTM2010-18318-E, MTM2011-29342 and Generalitat de Catalunya for funding under grant 2009SGR345. PG thanks the Wellcome Trust for support under grant 098325. This publication was based on work supported in part by Award No. KUK-013-04, made the King Abdullah University of Science and Technology (KAUST).

## Supplementary material

## References

- Aguirre-Ghiso JA (2007) Models, mechanisms and clinical evidence for cancer dormancy. Nat Rev Cancer 7:834–846CrossRefGoogle Scholar
- Alarcón T, Byrne HM, Maini PK (2004) A mathematical model of the effect of hypoxia on the cell-cycle of normal and cancer cells. J Theor Biol 229:395–411CrossRefGoogle Scholar
- Alarcón T, Page KM (2006) Stochastic models of receptor oligomerisation by bivalent ligand. J R Soc Interface 3:545–559CrossRefGoogle Scholar
- Alarcón T, Page KM (2007) Mathematical models of the VEGF receptor and its role in cancer therapy. J R Soc Interface 4:283–304CrossRefGoogle Scholar
- Alarcón T, Jensen HJ (2010) Quiescence: a mechanism for escaping the effects of drug on cell populations. J R Soc Interface 8:99–106CrossRefGoogle Scholar
- Becskei A, Serrano L (2000) Engineering stability in gene networks by autoregulation. Nature 405:590–593CrossRefGoogle Scholar
- Bedessem B, Stéphanou A (2014) A mathematical model of HIF-1-\(\alpha \)-mediated response to hypoxia on the G1/S transition. Math Biosci 248:31–39MathSciNetCrossRefzbMATHGoogle Scholar
- Blythe RA, McKane AJ (2007) Stochastic of evolution in genetics, ecology and linguistics. J Stat Mech P07018. doi: 10.1088/1742-5468/2007/07/P07018
- Bruna M, Chapman SJ (2012) Excluded-volume effects in the diffusion of hard spheres. Phys Rev E 85:011103CrossRefGoogle Scholar
- Chern Y, Cairns R, Papandreou I, Koong A, Denko NC (2009) Oxygen consumption can regulate the growth of tumours. A new perspective on the Warburg effect. PLoS One 4:e7033CrossRefGoogle Scholar
- Demetrius L, Gundlach VM, Ochs G (2009) Invasion exponents in biological networks. Physica A 388:651–672CrossRefGoogle Scholar
- Elowitz MB, Levine AJ, Siggia ED, Swain PS (2002) Stochastic gene expression in a single cell. Science 297:1183–1186CrossRefGoogle Scholar
- Enderling H, Almog N, Hlatky L (eds) (2012) Systems biology of tumour dormancy. Springer-Verlag, New YorkGoogle Scholar
- Escudero C, Kamenev A (2009) Switching rates in multistep reactions. Phys Rev E 79:041149CrossRefGoogle Scholar
- Ferrel JE, Xiong W (2001) Bistability in cell signalling: how to make continuous processes discontinuous, and reversible processes irreversible. Chaos 11:227–236CrossRefGoogle Scholar
- Gardiner CW (1983) The escape time in nonpotential systems. J Stat Phys 30:157–177CrossRefGoogle Scholar
- Gardiner CW (2009) Stochastic methods. Springer-Verlag, BerlinzbMATHGoogle Scholar
- Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22:403–434MathSciNetCrossRefGoogle Scholar
- Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361CrossRefGoogle Scholar
- Golbeter A, Koshland DE (1984) Ultrasensitivity in biochemical systems controlled by covalent modification. Interplay between zero-order and multistep effects. J Biol Chem 259:14441–14447Google Scholar
- Guerrero P, Alarcón T (2015) Stochastic multiscale models of cell populations: asymptotic and numerical methods. Math Model Nat Phen 10:64–93CrossRefGoogle Scholar
- Grimmett GR, Stirzaker DR (1992) Probability and random processes. Oxford University Press, OxfordGoogle Scholar
- Hanggi P, Talkner P, Borkovec M (1990) Reaction rate theory: 50 years after Kramers. Rev Mod Phys 62:251–341MathSciNetCrossRefGoogle Scholar
- Holte JM (1982) Critical multi-type branching processes. Ann Probab 10:482–495MathSciNetCrossRefzbMATHGoogle Scholar
- Horsthemke W, Lefever R (2006) Noise-induced transitions. Springer-Verlag, New YorkGoogle Scholar
- Hsu C, Scherrer S, Buetti-Dinh A, Ratna P, Pizzolato J, Jaquet V, Becskei A (2012) Stochastic signalling rewires the interaction map of multiple feedback network during yeast evolution. Nat Commun 3:682CrossRefGoogle Scholar
- Kelemen J, Ratna P, Scherrer S, Becskei A (2010) Spatial epigenetic control of mono- and bistable gene expression. PLoS Biol 8:e1000332CrossRefGoogle Scholar
- Kholodenko BN (2000) Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades. Eur J Biochem 267:1583–1588CrossRefGoogle Scholar
- Kimura M (1968) Evolutionary rate at the molecular level. Nature 217:624–626CrossRefGoogle Scholar
- Kimmel M, Axelrod DE (2002) Branching processes in biology. Springer-Verlag, New YorkCrossRefzbMATHGoogle Scholar
- Klausmeier CA (2008) Floquet theory: a useful tool for understanding non-equilibrium dynamics. Theor Ecol 1:153–161CrossRefGoogle Scholar
- Kitano H (2004) Cancer as a robust system: implications for cancer therapy. Nat Rev Cancer 4:227–235CrossRefGoogle Scholar
- Kubo R, Matsuo K, Kitahara K (1973) Fluctuation and relaxation of macrovariables. J Stat Phys 9:51–96CrossRefGoogle Scholar
- Ladbury JE, Arold ST (2012) Noise in cellular signalling pathways: causes and effects. Trends Biochem Sci 37:173–178CrossRefGoogle Scholar
- Legewie S, Blüthgen N, Herzel H (2006) Mathematical modelling identifies inhibitors of apoptosis as mediators of positive feed-back and bistability. PLoS Comput Biol 2:e120CrossRefGoogle Scholar
- Legewie S, Blüthgen N, Herzel H (2007) Competing docking interactions can bring about bistability in the MAPK cascade. Biophys J 93:2279–2288CrossRefGoogle Scholar
- Lestas I, Vinnicombe G, Paulsson J (2010) Fundamental limits on the suppression of molecular fluctuations. Nature 467:174–178CrossRefGoogle Scholar
- Lugo C, McKane AJ (2008) Quasicycles in a spatial predator–prey model. Phys Rev E 78:051911MathSciNetCrossRefGoogle Scholar
- Maier RS, Stein DL (1996) A scaling theory of bifurcations in the symmetric weak-noise escape problem. J Stat Phys 83:291–357MathSciNetCrossRefzbMATHGoogle Scholar
- Metz JAJ, Nisbet RM, Geritz SAH (1992) How should we define “fitness” for general ecological scenarios? Trends Ecol Evol 7:198–202CrossRefGoogle Scholar
- Munoz MA, Grinstein G, Tu Y (1997) Survival probability and field theory in systems with absorving states. Phys Rev E 56:5101–5105CrossRefGoogle Scholar
- Ortega F, Garcés JL, Mas F, Kholodenko BN, Cascante M (2006) Bistability from double phosphorylation in signal transduction. Kinetic and structural requirements. FEBS J 273:3915–3926CrossRefGoogle Scholar
- Rand DA, Wilson HB, McGlade JM (1994) Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics. Philos Trans R Soc Lond B 343:261–283CrossRefGoogle Scholar
- Rong L, Perelson AS (2009) Modelling latently infected cell activation: viral and latent reservoir persistence, and viral blips in HIV-infected patients on potent therapy. PLoS Comput Biol 5:e1000533MathSciNetCrossRefGoogle Scholar
- Strogatz SH (1994) Nonlinear dynamics and chaos. Perseus Books, New YorkGoogle Scholar
- Tian T, Olson S, Whitacre JM, Harding A (2011) The origin of cancer robustness and evolvability. Integr Biol 3:17–30CrossRefGoogle Scholar
- Touchette H (2009) The large deviation approach to statistical mechanics. Phys Rep 479:1–69MathSciNetCrossRefGoogle Scholar
- Tyson JJ, Novak B (2001) Regulation of the eukaryotic cell cycle: molecular antagonism, hysteresis, and irreversible trasitions. J Theor Biol 210:249–263CrossRefGoogle Scholar
- Tyson JJ, Chen KC, Novak B (2003) Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signalling pathways in the cell. Curr Opin Cell Biol 15:221–231CrossRefGoogle Scholar
- Wells A, Griffith L, Wells JZ, Taylor DP (2013) The dormancy dilemma: quiescence versus balanced proliferation. Cancer Res 73:3811–3816CrossRefGoogle Scholar
- Willis L, Alarcón T, Elia G, Jones JL, Wright N, Graham TA, Tomlinson IPM, Page KM (2010) Breast cancer dormancy can be maintained by a small number of micrometastases. Cancer Res 70:4310–4317CrossRefGoogle Scholar