We can thus represent a cell state by its expression pattern of the n genes where is the expression activity of gene locus quantified at the level of the genomic locus, either in the form of transcripts or proteins. The basins of attraction compartmentalize the network's state space and give rise to disjoint stable states — capturing essential properties of cell types . Three cell states with distinct gene expression and. The gene regulatory circuit of X and Y determines three cell states.
Each state is associated with a growth rate respectively. Three states transition to each other with the interconversion rates.
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A cell is the elementary unit in a population whose birth, death and transformation events underlie the population dynamics. Many studies describe the cellular transition using a master equation either in the discrete formalism, like Boolean networks  ,  , or in the continuous formalism of ordinary differentiation equations ODEs  — .
The assumption of mass conservation is generally used in models inspired by rate equations in chemistry. However, it needs to be taken into account that cellular multiplication violates the mass conservation. This notion is of central importance to understand tissue formation since the cell population dynamics become non-equilibrium dynamics.
The ratio between fractions of cells corresponding to different phenotypes no longer unconditionally approaches a steady state, considering both cell proliferation and cell transition. Together with the transition rate, the net cell growth proliferation minus death also changes the abundance of cells in attractor state and consequently affects the occupied ratio of attractor states, changing the overall tissue conformation.
These classical evolution models of cell populations have played an important role in the analysis of the somatic evolution of cancer cells, thought to be the major driver of cancer progression  , . However, these models tacitly assume a one-to-one mapping between genotype and phenotype and assume random genetic mutations as the mechanism for cell phenotype switching.
Recent advances in mammalian cell reprogramming and cell transdifferentiation have underscored the importance of multistability and non-genetic cell state transitions resulting in non-genetic cell population dynamics  , . Considering such non-genetic dynamics will lead to models that differ from classical population genetics models in the following points:. The key focus of this paper is the dynamics of the cellular composition of a growing cancer cell population. More specifically, we study the dynamics of the relative abundance of distinct cancer cell subtypes.
We also discuss the conditions for a cancer cell population to maintain the fixed ratio and distinct cell types. Finally we study how transition and growth rates influence the subpopulation ratio when cell population reaches equilibrium. Since the true novelty of this research is to introduce the state-dependent growth rate to cell state transition, in general this model can be used to describe the cell differentiation during embryogenesis, or any cell population dynamics in which growth and transition both play important roles in the same scale.
We start with a simple model of cancer cell population with two-phenotypes. We assume that bimodal expression levels of markers can modulate the growth rate. The discretization of a cell population with continuously distributed gene expression levels into two states, and , is appropriated to capture the characteristic population dynamics.
Nonlinear dynamical systems usually have stable steady states attractors and the system should return to attractors under reasonably small perturbations. The dynamics of returning to the attractors or re-establishing the equilibrium after perturbation is usually a key property of nonlinear dynamical systems.
It helps us to understand the mechanism of many interesting biological phenomena in biology, such as polymorphism, homeostasis etc. Here we first establish the conditions for the co-existence of the two phenotypes by characterizing both the existence of a steady-state ratio of these two types and the dynamics underpinning the re-establishment of the equilibrium.
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The net growth rates the quantities readily measured are and , respectively. The cell population dynamics including both cell growth and state transition are described by the corresponding set of ODEs: 1. It is important to emphasize that only the net growth rate can be reliably measured in cell culture. If are eigenvalues of in Eq.
The long-term dynamic behavior of this dynamic system is then determined by the two eigenvalues. There are two distinct behaviors if the eigenvalues either have opposite signs or the same sign. When the two eigenvalues have opposite signs, i. In this situation the two cell populations have essentially the same growth rate, defined by the positive eigenvalue with only different pre-factors A 11 when and A 22 when.
This indicates that only one subpopulation can survive independently while the second subpopulation lives as the derivative of the other one. When the two eigenvalues have the same signs, i.
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If both eigenvalues are positive, the two subpopulations survive together. In the long term both populations will grow with the same rate, defined by the larger eigenvalue. Finally, if both eigenvalues are negative, then both subpopulations become extinct together.
Mathematically, since the product of the two eigenvalues is the determinant of the matrix T , the condition for the two eigenvalues having the same sign can be written as: 4. Therefore, if the growth rates for both phenotypes are much greater than the state conversions, i. This simple mathematical observation has consequences in non-genetic drug resistance persistors  , .
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If the conversion rates of both subpopulations are much larger than their respective growth rates, the distinction of the two discrete cell phenotypes becomes blurred. Interestingly, in this case none of the subpopulations can survive alone. We can consider these two populations as a single one with a mean growth rate of. Thus, by examining transition rates in regimes never considered in mutation-based population dynamics because mutations are rare , we enter a dynamical regime that is relevant for cell population dynamics in which non-genetic phenotype conversions dominate.
We can determine the population ratio between the two subpopulations in this regime. Since in the long term both populations growth is given by the term with the larger eigenvalue, i. The dynamics of follow 5. Therefore, the steady-state ratio fraction of is 6 in which we denote as the differential net growth rate of with respect to that of divided by the transition rate from to.
While the population ratio becomes stationary, both populations of and can grow indefinitely. This result deviates from classical population dynamics in which the coexistence of two populations of different growth rates is not stable due to one-direction conversion mutation. In fact, recent work on clonal isogenic cancer cell populations showed that they typically consist of several interconverting discrete sub-populations associated with biologically relevant functional properties, such as stem-like behavior, drug—efflux capability  ,  and differentiation .
The question now is: Can we quantify the different influences on the observed cell fixed ratio from the growth and transition rates?
A possible biological interpretation is that changes in and relative to each other represent differential fitness in a given environment, which could promote Darwinian selection. Along the same line, changes in can represent Lamarckian instruction in the sense that a given environment may impose differential transition rates between different phenotypes. This offers a simple mathematical framework to describe the relative contribution of Darwinian selection and Lamarckian instruction in shifting population ratios during tumor progression under chemotherapy.
It is easy to obtain the time-course for the dynamics of the re-equilibrium of the subpopulations by finding the integral solution for Eq. For the convenience of integration, we change the variable to and integrate it from the initial cell population ratio to the ratio at any arbitrary time : 8. Since , where is given by Eq. This result suggests that the re-equilibrium time is in the order of.
We can also benchmark this equation with two extreme cases whose transition rates are obvious. The initial rate for re-equilibrium starting with a pure subpopulation , i. In the general case, re-equilibrium rate is a dynamic process that combines both Eq. The right-hand-side of Eq.
Therefore, if , i. The rate for re-equilibrium increases with time until , followed by a decreasing rate. This is shown in Fig. However, if , i. In this case, the time course of re-equilibration follows a exponential saturation kinetics monotonically increasing before reaching the saturation , as shown in Fig. Cell differential growth rates are bigger than cell differential transition rates.
The time derivative of cell ratio is non-monotonic before reaching the fixed ratio. Cell differential growth rates are smaller than the cell differential transition rate. The time derivative of cell ratio is monotonically decreasing before reaching the fixed ratio.
Cell re-equilibrium dynamics is sigmoidal for the condition shown in A. Cell re-equilibrium dynamics is logistical for the condition shown in B.
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During the development of multicellular organisms, usually more than two cell types are formed. This introduces qualitatively new properties not seen in the classical two-state transition model. To study this phenomenon, we extended our mathematical formalism above to cell phenotypes: Here are the respective number of cells of phenotypes in the cell population.
Each phenotype has a respective growth rate ; are the state transition rates from cell phenotype i to phenotype j. Since this is a linear system, the right-hand-side can be decomposed as a sum of a diagonal matrix and a Markov matrix. The matrix form of Eq. Here is a positive diagonal matrix with growth rates and is a Markov matrix consisting of transition rates.