## 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.

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## References

Aguirre-Ghiso JA (2007) Models, mechanisms and clinical evidence for cancer dormancy. Nat Rev Cancer 7:834–846

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 229:395–411

Alarcón T, Page KM (2006) Stochastic models of receptor oligomerisation by bivalent ligand. J R Soc Interface 3:545–559

Alarcón T, Page KM (2007) Mathematical models of the VEGF receptor and its role in cancer therapy. J R Soc Interface 4:283–304

Alarcón T, Jensen HJ (2010) Quiescence: a mechanism for escaping the effects of drug on cell populations. J R Soc Interface 8:99–106

Becskei A, Serrano L (2000) Engineering stability in gene networks by autoregulation. Nature 405:590–593

Bedessem B, Stéphanou A (2014) A mathematical model of HIF-1-\(\alpha \)-mediated response to hypoxia on the G1/S transition. Math Biosci 248:31–39

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 85:011103

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:e7033

Demetrius L, Gundlach VM, Ochs G (2009) Invasion exponents in biological networks. Physica A 388:651–672

Elowitz MB, Levine AJ, Siggia ED, Swain PS (2002) Stochastic gene expression in a single cell. Science 297:1183–1186

Enderling H, Almog N, Hlatky L (eds) (2012) Systems biology of tumour dormancy. Springer-Verlag, New York

Escudero C, Kamenev A (2009) Switching rates in multistep reactions. Phys Rev E 79:041149

Ferrel JE, Xiong W (2001) Bistability in cell signalling: how to make continuous processes discontinuous, and reversible processes irreversible. Chaos 11:227–236

Gardiner CW (1983) The escape time in nonpotential systems. J Stat Phys 30:157–177

Gardiner CW (2009) Stochastic methods. Springer-Verlag, Berlin

Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22:403–434

Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361

Golbeter A, Koshland DE (1984) Ultrasensitivity in biochemical systems controlled by covalent modification. Interplay between zero-order and multistep effects. J Biol Chem 259:14441–14447

Guerrero P, Alarcón T (2015) Stochastic multiscale models of cell populations: asymptotic and numerical methods. Math Model Nat Phen 10:64–93

Grimmett GR, Stirzaker DR (1992) Probability and random processes. Oxford University Press, Oxford

Hanggi P, Talkner P, Borkovec M (1990) Reaction rate theory: 50 years after Kramers. Rev Mod Phys 62:251–341

Holte JM (1982) Critical multi-type branching processes. Ann Probab 10:482–495

Horsthemke W, Lefever R (2006) Noise-induced transitions. Springer-Verlag, New York

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 3:682

Kelemen J, Ratna P, Scherrer S, Becskei A (2010) Spatial epigenetic control of mono- and bistable gene expression. PLoS Biol 8:e1000332

Kholodenko BN (2000) Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades. Eur J Biochem 267:1583–1588

Kimura M (1968) Evolutionary rate at the molecular level. Nature 217:624–626

Kimmel M, Axelrod DE (2002) Branching processes in biology. Springer-Verlag, New York

Klausmeier CA (2008) Floquet theory: a useful tool for understanding non-equilibrium dynamics. Theor Ecol 1:153–161

Kitano H (2004) Cancer as a robust system: implications for cancer therapy. Nat Rev Cancer 4:227–235

Kubo R, Matsuo K, Kitahara K (1973) Fluctuation and relaxation of macrovariables. J Stat Phys 9:51–96

Ladbury JE, Arold ST (2012) Noise in cellular signalling pathways: causes and effects. Trends Biochem Sci 37:173–178

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:e120

Legewie S, Blüthgen N, Herzel H (2007) Competing docking interactions can bring about bistability in the MAPK cascade. Biophys J 93:2279–2288

Lestas I, Vinnicombe G, Paulsson J (2010) Fundamental limits on the suppression of molecular fluctuations. Nature 467:174–178

Lugo C, McKane AJ (2008) Quasicycles in a spatial predator–prey model. Phys Rev E 78:051911

Maier RS, Stein DL (1996) A scaling theory of bifurcations in the symmetric weak-noise escape problem. J Stat Phys 83:291–357

Metz JAJ, Nisbet RM, Geritz SAH (1992) How should we define “fitness” for general ecological scenarios? Trends Ecol Evol 7:198–202

Munoz MA, Grinstein G, Tu Y (1997) Survival probability and field theory in systems with absorving states. Phys Rev E 56:5101–5105

Ortega F, Garcés JL, Mas F, Kholodenko BN, Cascante M (2006) Bistability from double phosphorylation in signal transduction. Kinetic and structural requirements. FEBS J 273:3915–3926

Rand DA, Wilson HB, McGlade JM (1994) Dynamics and evolution: evolutionarily stable attractors, invasion exponents and phenotype dynamics. Philos Trans R Soc Lond B 343:261–283

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:e1000533

Strogatz SH (1994) Nonlinear dynamics and chaos. Perseus Books, New York

Tian T, Olson S, Whitacre JM, Harding A (2011) The origin of cancer robustness and evolvability. Integr Biol 3:17–30

Touchette H (2009) The large deviation approach to statistical mechanics. Phys Rep 479:1–69

Tyson JJ, Novak B (2001) Regulation of the eukaryotic cell cycle: molecular antagonism, hysteresis, and irreversible trasitions. J Theor Biol 210:249–263

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 15:221–231

Wells A, Griffith L, Wells JZ, Taylor DP (2013) The dormancy dilemma: quiescence versus balanced proliferation. Cancer Res 73:3811–3816

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 70:4310–4317

## 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).

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## Appendix: Noise in bistable systems

### Appendix: Noise in bistable systems

This appendix is devoted to a short summary of the effect of noise in bistable systems. Our aim is not to give a full account of stochastic dynamics of bistable systems, but to provide a brief justification of why phenotype switching in the bistable population can be modelled as an activated process and, consequently, the corresponding transition rates as functions of the Arrhenius type. For a full account, we refer the reader to the the extensive literature on the subject, in particular Gardiner (1983), Maier and Stein (1996), Horsthemke and Lefever (2006), Gardiner (2009).

For concreteness, we consider the following dynamical system:

This system was proposed by Tyson and Novak (2001) as the central component of a more complex pathway regulating the G\(_1\)/S transition in the cell-cycle of eukaryote cells. In their original model, \(x\) stands for the (normalised) concentration of active Cdh1/APC complexes, \(y\) for the concentration of active CycB/CDK complexes, and \(m\) for the cell size. Here, \(m\) will be assumed to be the control parameter.

As \(m\) increases, Eqs. (36) go through a series of bifurcations which separate different regimes (see Fig. 14): For small values of \(m\) the system exhibits a single steady-state (high \(x=\) Cdh1, low \(y=\) CycB corresponding to G\(_1\)). Intermediate values of \(m\) lead to a bistable regime where two stable steady-states (high Cdh1, low CycB corresponding to G\(_1\) and low Cdh1 and high CycB, corresponding to S–G\(_2\)–M) coexist with a saddle point. Finally, as \(m\) continues to increase a saddle-node bifurcation occurs which leads to the annihilation of the G\(_1\)-like steady-state and the saddle point. In each of the panels of Fig. 14, we show two solutions of Eqs. (36) corresponding to two different initial conditions: \(y(t=0)=0.28\) and \(x(t=0)=0.9\) (solid green lines) and \(y(t=0)=0.3\) and \(x(t=0)=0.9\) (solid purple lines). We have also plotted realisations of a stochastic system equivalent to Eqs. (36) (see Guerrero and Alarcón (2015) for details) with the same initial conditions (dashed green lines and dashed purple lines, respectively). In the two mono-stable cases shown in panels \(m=0.01\) and \(m=1\), we see that there are no major differences in behaviour between the mean-field (solid lines) and the stochastic (dashed lines) systems: In both cases the stochastic trajectories converge toward the mean-field fixed point, regardless of the initial condition. The behaviour in the bistable regime offers more possibilities. We have chosen the initial conditions (for both the mean-field equations and the stochastic system) so that they belong to different basins of attraction of the mean-field system: \(y(t=0)=0.28\) and \(x(t=0)=0.9\) (solid green lines) belongs to that of G\(_1\), \(y(t=0)=0.3\) and \(x(t=0)=0.9\) (solid purple lines), to that of S-G\(_2\)-M the bistable regime (\(m=0.2\)). In this regime, we can see that the stochastic trajectories (dashed lines) may either behave like their mean-field counterparts and converge towards the corresponding steady-state, or, on the contrary, jump across the separatrix and converge towards the steady-state corresponding to the other (mean-field) basin of attraction.

Let us focus now on the bistable regime (e.g. \(m=0.2\) in Fig. 14). If noise is ignored, then the stable steady-states have disjoint basins of attraction, being separated by a separatrix which passes through the saddle: If the initial condition is contained in the \(G_1\)-basin (respectively, S-G\(_2\)-M) then the system will evolve towards \(G_1\) (respectively, S-G\(_2\)-M). By contrast, if noise is taken into account then the separatrix becomes a barrier that the system may cross with finite probability. This is shown in Fig. 14 where we compare the solution of Eqs. (36) with two different initial conditions, one on each side of the separatrix, with a stochastic version of the Tyson & Novak model developed in Guerrero and Alarcón (2015).

There is extensive literature Escudero and Kamenev (2009), Gardiner (1983), Gardiner (2009), Horsthemke and Lefever (2006), Maier and Stein (1996) showing that, in the limit of low noise intensity, \(\sigma \), the process of switching between two stable states is an activated process, i.e. the transition rate from one steady state to the other, \(W_T\), is such that \(W_T\sim e^{-H}\) for \(\sigma \ll 1\) where \(H\) is a function of the parameters of the system.

While a detailed mathematical derivation is beyond the scope of this appendix, the physical rationale is relatively straightforward Kubo et al. (1973). In equilibrium statistical mechanics, the statistical distribution for an extensive variable, say \(X\), is given by \(P_e(X)=Z^{-1}e^{-\frac{{V}}{k_BT}\phi _e(x)}\), where \({V}\) is the volume of the system, \(x=X/{V}\), \(k_BT\) is the energy associated with thermal noise (\(T\) is the temperature and \(k_B\) is Boltzmann’s constant), and \(\phi _e(x)\) is the equilibrium free energy per unit volume. From elementary considerations in equilibrium thermodynamics, we know that the equilibrium value of \(x\) corresponds to the minimum of \(\phi _e(x)\) which, in turn, corresponds to the most probable value of \(x\) according to the probability distribution \(P_e(X)\), with \(P_e(X)\) providing the probability of deviation of \(x\) from its *optimal* value, \(x_e\). Incidentally, we remark that for low noise intensity, i.e. for \(k_BT\) much smaller than a characteristic energy scale associated with \(\phi _e(x)\), large deviations from \(x_e\) are very unlikely as they are exponentially suppressed Touchette (2009). We note also that the larger the size of the system, \({V}\), the more insensitive the system becomes to random effects.

The essential ansatz leading to \(W_T\sim e^{-H}\) is that non-equilibrium (i.e. time-evolving) states of *large* systems can be described by a probability distribution which is a straightforward generalisation of \(P_e\), i.e.:

where \(x(t)=X(t)/{V}\) is a sample path or realisation of the stochastic process, \(x_0=x(0)\) denotes the initial condition and \(\sigma \) is the intensity of the noise (which plays the same role as the temperature \(k_BT\) in the equilibrium case). Therefore, the most likely evolution for \(x(t)\) is along the path that minimises \(\phi (x(t)\vert x_0)\), and \(P(X(t)\vert X_0)\) can, again, be understood as the probability that a sample path deviates from the optimal (most probable) path, \(x_o(t)\). For large systems, \(\varOmega \gg 1\), such deviations are very unlikely (they are exponentially suppressed) and the most significant contribution comes from \(x_o(t)\). Hence, provided the noise \(\sigma \equiv {V}^{-1}\ll 1\), the transition probability of the system to start from \(X_0\) and evolve to a state \(X_1\) at time \(t\), \(W_T\), can be estimated by taking \(\phi =\phi _0\equiv \phi (x_o(t)\vert x_0)\) in Eq. (37), namely, \(W_T\sim e^{-H}\) where \(H\equiv \frac{{V}}{\sigma }\phi _0\).

Although we have presented this argument in an heuristic (non-rigorous) way, it can be made mathematically rigorous within the framework of the theory of large deviations Kubo et al. (1973), Touchette (2009).

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Guerrero, P., Byrne, H.M., Maini, P.K. *et al.* From invasion to latency: intracellular noise and cell motility as key controls of the competition between resource-limited cellular populations.
*J. Math. Biol.* **72**, 123–156 (2016). https://doi.org/10.1007/s00285-015-0883-2

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DOI: https://doi.org/10.1007/s00285-015-0883-2