Fitness and entropy production in a cell population dynamics with epigenetic phenotype switching

Motivated by recent understandings in the stochastic natures of gene expression, biochemical signaling, and spontaneous reversible epigenetic switchings, we study a simple deterministic cell population dynamics in which subpopulations grow with different rates and individual cells can bi-directionally switch between a small number of different epigenetic phenotypes. Two theories in the past, the population dynamics and thermodynamics of master equations, separately defined two important concepts in mathematical terms: the fitness in the former and the (non-adiabatic) entropy production in the latter. Both of them play important roles in the evolution of the cell population dynamics. The switching sustains the variations among the subpopulation growth, thus sustains continuous natural selection. As a form of Price’s equation, the fitness increases with (i) natural selection through variations and (ii) a positive covariance between the per capita growth and switching, which represents a Lamarchian-like behavior. A negative covariance balances the natural selection in a fitness steady state --- “the red queen” scenario. At the same time the growth keeps the proportions of subpopulations away from the “intrinsic” switching equilibrium of individual cells, thus leads to a continuous entropy production. A covariance, between the per capita growth rate and the “chemical potential” of subpopulation, counteracts the entropy production. Analytical results are obtained for the limiting cases of growth dominating switching and vice versa.     

Stochastic gene expression in fluctuating environments

Stochastic mechanisms can cause a group of isogenic bacteria, each subject to identical environmental conditions, to nevertheless exhibit diverse patterns of gene expression. The resulting phenotypic subpopulations will typically have distinct growth rates. This behavior has been observed in several contexts, including sugar metabolism and pili phase variation. Under fixed environmental conditions, the net growth rate of the population is maximized when all cells are of the fastest growing phenotype, so it is unclear what fitness advantage is conferred by population heterogeneity. However, unlike ideal laboratory conditions, natural environments tend to fluctuate, either periodically or randomly. Here we use a stochastic population model to show that, during growth in such fluctuating environments, a dynamically heterogenous bacterial population can sometimes achieve a higher net growth rate than a homogenous one. By using stochastic mechanisms to sample several distinct phenotypes, the bacteria are able to anticipate and take advantage of sudden changes in their environment. However, this heterogeneity is beneficial only if the bacterial response rate is sufficiently low. Our results could be useful in the design of artificial evolution experiments and in the optimization of fermentation processes.     

Bacterial persistence: a model of survival in changing environments

The persistence phenotype is an epigenetic trait exhibited by a subpopulation of bacteria, characterized by slow growth coupled with an ability to survive antibiotic treatment. The phenotype is acquired via a spontaneous, reversible switch between normal and persister cells. These observations suggest that clonal bacterial populations may use persister cells, whose slow division rate under growth conditions leads to lower population fitness, as an "insurance policy" against antibiotic encounters. We present a model of Escherichia coli persistence, and using experimentally derived parameters for both wild type and a mutant strain (hipQ) with markedly different switching rates, we show how fitness loss due to slow persister growth pays off as a risk-reducing strategy. We demonstrate that wild-type persistence is suited for environments in which antibiotic stress is a rare event. The optimal rate of switching between normal and persister cells is found to depend strongly on the frequency of environmental changes and only weakly on the selective pressures of any given environment. In contrast to typical examples of adaptations to features of a single environment, persistence appears to constitute an adaptation that is tuned to the distribution of environmental change.     

Evolution and Phenotypic Selection of Cancer Stem Cells

Cells of different organs at different ages have an intrinsic set of kinetics that dictates their behavior. Transformation into cancer cells will inherit these kinetics that determine initial cell and tumor population progression dynamics. Subject to genetic mutation and epigenetic alterations, cancer cell kinetics can change, and favorable alterations that increase cellular fitness will manifest themselves and accelerate tumor progression. We set out to investigate the emerging intratumoral heterogeneity and to determine the evolutionary trajectories of the combination of cell-intrinsic kinetics that yield aggressive tumor growth. We develop a cellular automaton model that tracks the temporal evolution of the malignant subpopulation of so-called cancer stem cells(CSC), as these cells are exclusively able to initiate and sustain tumors. We explore orthogonal cell traits, including cell migration to facilitate invasion, spontaneous cell death due to genetic drift after accumulation of irreversible deleterious mutations, symmetric cancer stem cell division that increases the cancer stem cell pool, and telomere length and erosion as a mitotic counter for inherited non-stem cancer cell proliferation potential. Our study suggests that cell proliferation potential is the strongest modulator of tumor growth. Early increase in proliferation potential yields larger populations of non-stem cancer cells(CC) that compete with CSC and thus inhibit CSC division while a reduction in proliferation potential loosens such inhibition and facilitates frequent CSC division. The sub-population of cancer stem cells in itself becomes highly heterogeneous dictating population level dynamics that vary from long-term dormancy to aggressive progression. Our study suggests that the clonal diversity that is captured in single tumor biopsy samples represents only a small proportion of the total number of phenotypes.     

Source-sink dynamics shape the evolution of antibiotic resistance and its pleiotropic fitness cost

Understanding the conditions that favour the evolution and maintenance of antibiotic resistance is the central goal of epidemiology. A crucial feature explaining the adaptation to harsh, or 'sink', environments is the supply of beneficial mutations via migration from a 'source' population. Given that antibiotic resistance is frequently associated with antagonistic pleiotropic fitness costs, increased migration rate is predicted not only to increase the rate of resistance evolution but also to increase the probability of fixation of resistance mutations with minimal fitness costs. Here we report in vitro experiments using the nosocomial pathogenic bacterium Pseudomonas aeruginosa that support these predictions: increasing rate of migration into environments containing antibiotics increased the rate of resistance evolution and decreased the associated costs of resistance. Consistent with previous theoretical work, we found that resistance evolution arose more rapidly in the presence of a single antibiotic than two. Evolution of resistance was also more rapid when bacteria were subjected to sequential exposure with two antibiotics (cycling therapy) compared with simultaneous exposure (bi-therapy). Furthermore, pleiotropic fitness costs of resistance to two antibiotics were higher than for one antibiotic, and were also higher under bi-therapy than cycling therapy, although the cost of resistance depended on the order of the antibiotics through time. These results may be relevant to the clinical setting where immigration is known to be important between chemotherapeutically treated patients, and demonstrate the importance of ecological and evolutionary dynamics in the control of antibiotic resistance.     

Stochastic and deterministic simulations of heterogeneous cell population dynamics

A Monte Carlo algorithm, which can accurately simulate the dynamics of entire heterogeneous cell populations, was developed. The algorithm takes into account the random nature of cell division as well as unequal partitioning of cellular material at cell division. Moreover, it is general in the sense that it can accommodate a variety of single-cell, deterministic reaction kinetics as well as various stochastic division and partitioning mechanisms. The validity of the algorithm was assessed through comparison of its results with those of the corresponding deterministic cell population balance model in cases where stochastic behavior is expected to be quantitatively negligible. Both algorithms were applied to study: (a) linear intracellular kinetics and (b) the expression dynamics of a genetic network with positive feedback architecture, such as the lac operon. The effects of stochastic division as well as those of different division and partitioning mechanisms were assessed in these systems, while the comparison of the stochastic model with a continuum model elucidated the significance of cell population heterogeneity even in cases where only the prediction of average properties is of primary interest.     

Compensatory evolution of pbp mutations restores the fitness cost imposed by β-lactam resistance in Streptococcus pneumoniae

The prevalence of antibiotic resistance genes in pathogenic bacteria is a major challenge to treating many infectious diseases. The spread of these genes is driven by the strong selection imposed by the use of antibacterial drugs. However, in the absence of drug selection, antibiotic resistance genes impose a fitness cost, which can be ameliorated by compensatory mutations. In Streptococcus pneumoniae, β-lactam resistance is caused by mutations in three penicillin-binding proteins, PBP1a, PBP2x, and PBP2b, all of which are implicated in cell wall synthesis and the cell division cycle. We found that the fitness cost and cell division defects conferred by pbp2b mutations (as determined by fitness competitive assays in vitro and in vivo and fluorescence microscopy) were fully compensated by the acquisition of pbp2x and pbp1a mutations, apparently by means of an increased stability and a consequent mislocalization of these protein mutants. Thus, these compensatory combinations of pbp mutant alleles resulted in an increase in the level and spectrum of β-lactam resistance. This report describes a direct correlation between antibiotic resistance increase and fitness cost compensation, both caused by the same gene mutations acquired by horizontal transfer. The clinical origin of the pbp mutations suggests that this intergenic compensatory process is involved in the persistence of β-lactam resistance among circulating strains. We propose that this compensatory mechanism is relevant for β-lactam resistance evolution in Streptococcus pneumoniae.     

Phenotypic Heterogeneity,a Phenomenon That May Explain Why Quorum Sensing Does Not Always Result in Truly Homogenous Cell Behavior

Phenotypic heterogeneity describes the occurrence of “nonconformist” cells within an isogenic population. The nonconformists show an expression profile partially different from that of the remainder of the population. Phenotypic heterogeneity affects many aspects of the different bacterial lifestyles, and it is assumed that it increases bacterial fitness and the chances for survival of the whole population or smaller subpopulations in unfavorable environments. Well-known examples for phenotypic heterogeneity have been associated with antibiotic resistance and frequently occurring persister cells. Other examples include heterogeneous behavior within biofilms, DNA uptake and bacterial competence, motility (i.e., the synthesis of additional flagella), onset of spore formation, lysis of phages within a small subpopulation, and others. Interestingly, phenotypic heterogeneity was recently also observed with respect to quorum-sensing (QS)-dependent processes, and the expression of autoinducer (AI) synthase genes and other QS-dependent genes was found to be highly heterogeneous at a single-cell level. This phenomenon was observed in several Gram-negative bacteria affiliated with the genera Vibrio, Dinoroseobacter, Pseudomonas, Sinorhizobium, and Mesorhizobium. A similar observation was made for the Gram-positive bacterium Listeria monocytogenes. Since AI molecules have historically been thought to be the keys to homogeneous behavior within isogenic populations, the observation of heterogeneous expression is quite intriguing and adds a new level of complexity to the QS-dependent regulatory networks. All together, the many examples of phenotypic heterogeneity imply that we may have to partially revise the concept of homogeneous and coordinated gene expression in isogenic bacterial populations.     

When to wake up? The optimal waking-up strategies for starvation-induced persistence

Prolonged lag time can be induced by starvation contributing to the antibiotic tolerance of bacteria. We analyze the optimal lag time to survive and grow the iterative and stochastic application of antibiotics. A simple model shows that the optimal lag time can exhibit a discontinuous transition when the severeness of the antibiotic application, such as the probability to be exposed the antibiotic, the death rate under the exposure, and the duration of the exposure, is increased. This suggests the possibility of reducing tolerant bacteria by controlled usage of antibiotics application. When the bacterial populations are able to have two phenotypes with different lag times, the fraction of the second phenotype that has different lag time shows a continuous transition. We then present a generic framework to investigate the optimal lag time distribution for total population fitness for a given distribution of the antibiotic application duration. The obtained optimal distributions have multiple peaks for a wide range of the antibiotic application duration distributions, including the case where the latter is monotonically decreasing. The analysis supports the advantage in evolving multiple, possibly discrete phenotypes in lag time for bacterial long-term fitness.     

Evolution of alternative insect life histories in stochastic seasonal environments

Deterministic seasonality can explain the evolution of alternative life history phenotypes (i.e., life history polyphenism) expressed in different generations emerging within the same year. However, the influence of stochastic variation on the expression of such life history polyphenisms in seasonal environments is insufficiently understood. Here, we use insects as a model and explore (1) the effects of stochastic variation in seasonality and (2) the life cycle on the degree of life history differentiation among the alternative developmental pathways of direct development and diapause (overwintering), and (3) the evolution of phenology. With numerical simulation, we determine the values of development (growth) time, growth rate, body size, reproductive effort, adult life span, and fecundity in both the overwintering and directly developing generations that maximize geometric mean fitness. The results suggest that natural selection favors the expression of alternative life histories in the alternative developmental pathways even when there is stochastic variation in seasonality, but that trait differentiation is affected by the developmental stage that overwinters. Increasing environmental unpredictability induced a switch to a bet‐hedging type of life history strategy, which is consistent with general life history theory. Bet‐hedging appeared in our study system as reduced expression of the direct development phenotype, with associated changes in life history phenotypes, because the fitness value of direct development is highly variable in uncertain environments. Our main result is that seasonality itself is a key factor promoting the evolution of seasonally polyphenic life histories but that environmental stochasticity may modulate the expression of life history phenotypes.     

Antibody‐mediated crosslinking of gut bacteria hinders the spread of antibiotic resistance

The body is home to a diverse microbiota, mainly in the gut. Resistant bacteria are selected by antibiotic treatments, and once resistance becomes widespread in a population of hosts, antibiotics become useless. Here, we develop a multiscale model of the interaction between antibiotic use and resistance spread in a host population, focusing on an important aspect of within‐host immunity. Antibodies secreted in the gut enchain bacteria upon division, yielding clonal clusters of bacteria. We demonstrate that immunity‐driven bacteria clustering can hinder the spread of a novel resistant bacterial strain in a host population. We quantify this effect both in the case where resistance preexists and in the case where acquiring a new resistance mutation is necessary for the bacteria to spread. We further show that the reduction of spread by clustering can be countered when immune hosts are silent carriers, and are less likely to get treated, and/or have more contacts. We demonstrate the robustness of our findings to including stochastic within‐host bacterial growth, a fitness cost of resistance, and its compensation. Our results highlight the importance of interactions between immunity and the spread of antibiotic resistance, and argue in the favor of vaccine‐based strategies to combat antibiotic resistance.     

Hidden hysteresis – population dynamics can obscure gene network dynamics

Background

Positive feedback is a common motif in gene regulatory networks. It can be used in synthetic networks as an amplifier to increase the level of gene expression, as well as a nonlinear module to create bistable gene networks that display hysteresis in response to a given stimulus. Using a synthetic positive feedback-based tetracycline sensor in E. coli, we show that the population dynamics of a cell culture has a profound effect on the observed hysteretic response of a population of cells with this synthetic gene circuit.

Results

The amount of observable hysteresis in a cell culture harboring the gene circuit depended on the initial concentration of cells within the culture. The magnitude of the hysteresis observed was inversely related to the dilution procedure used to inoculate the subcultures; the higher the dilution of the cell culture, lower was the observed hysteresis of that culture at steady state. Although the behavior of the gene circuit in individual cells did not change significantly in the different subcultures, the proportion of cells exhibiting high levels of steady-state gene expression did change.Although the interrelated kinetics of gene expression and cell growth are unpredictable at first sight, we were able to resolve the surprising dilution-dependent hysteresis as a result of two interrelated phenomena - the stochastic switching between the ON and OFF phenotypes that led to the cumulative failure of the gene circuit over time, and the nonlinear, logistic growth of the cell in the batch culture.

Conclusions

These findings reinforce the fact that population dynamics cannot be ignored in analyzing the dynamics of gene networks. Indeed population dynamics may play a significant role in the manifestation of bistability and hysteresis, and is an important consideration when designing synthetic gene circuits intended for long-term application.

     

Environmental Uncertainty and Variable Diapause

We analyze a stage-structured model of a population that displays variable diapause in a randomly varying environment. The ruggedness of the environment is measured by the extent of random variation in per-capita reproductive success. We show how variable diapause and environmental characteristics affect the population′s stochastic growth rate. In rugged unpredictable environments, phenotypes that show some tendency to diapause are found to have a higher growth rate than nondiapausing phenotypes. In harsh rugged environments, some tendency to diapause may be all that permits population persistence. Positive serial autocorrelation causes the optimal diapause fraction to decrease, while negative autocorrelation causes that fraction to increase. The structured model behaves very differently from a scalar model for large diapause fractions even in uncorrelated environments, and in many cases predicts a broad optimum. The difference between models is due to the extreme variability of stage structure in populations subject to even small variability when diapause tendency is high.     

Acquisition of resistance genes by the IncQ plasmid R1162 is limited by its high copy number and lack of a partitioning mechanism.

R1162 is a representative member of the broad-host-range IncQ group of multicopy plasmids. Lower-copy-number derivatives of R1162 were constructed in vitro and shown to be unstable, indicating that partitioning of plasmid copies at cell division is due to random distribution and not to an active partitioning mechanism. However, the normal copy number of R1162 reduces cell fitness during growth in broth and favors the emergence of unstable, lower-copy-number variants. As a result, plasmid-borne antibiotic resistance genes active at a low copy number eventually result in plasmid loss during periods of no selection. We argue that the maintenance of R1162 in a population requires a gene that is selected only at high levels. This reduces the potential for acquiring genes from other R factors and could explain the limited variety of antibiotic resistance genes contained by naturally occurring IncQ plasmids.     

Bistability in feedback circuits as a byproduct of evolution of evolvability

Noisy bistable dynamics in gene regulation can underlie stochastic switching and is demonstrated to be beneficial under fluctuating environments. It is not known, however, if fluctuating selection alone can result in bistable dynamics. Using a stochastic model of simple feedback networks, we apply fluctuating selection on gene expression and run in silico evolutionary simulations. We find that independent of the specific nature of the environment–fitness relationship, the main outcome of fluctuating selection is the evolution of increased evolvability in the network; system parameters evolve toward a nonlinear regime where phenotypic diversity is increased and small changes in genotype cause large changes in expression level. In the presence of noise, the evolution of increased nonlinearity results in the emergence and maintenance of bistability. Our results provide the first direct evidence that bistability and stochastic switching in a gene regulatory network can emerge as a mechanism to cope with fluctuating environments. They strongly suggest that such emergence occurs as a byproduct of evolution of evolvability and exploitation of noise by evolution.     

Expected relative fitness and the adaptive topography of fluctuating selection

Wright's adaptive topography describes gene frequency evolution as a maximization of mean fitness in a constant environment. I extended this to a fluctuating environment by unifying theories of stochastic demography and fluctuating selection, assuming small or moderate fluctuations in demographic rates with a stationary distribution, and weak selection among the types. The demography of a large population, composed of haploid genotypes at a single locus or normally distributed phenotypes, can then be approximated as a diffusion process and transformed to produce the dynamics of population size, N, and gene frequency, p, or mean phenotype, . The expected evolution of p or is a product of genetic variability and the gradient of the long-run growth rate of the population, , with respect to p or . This shows that the expected evolution maximizes , the mean Malthusian fitness in the average environment minus half the environmental variance in population growth rate. Thus, as a function of p or represents an adaptive topography that, despite environmental fluctuations, does not change with time. The haploid model is dominated by environmental stochasticity, so the expected maximization is not realized. Different constraints on quantitative genetic variability, and stabilizing selection in the average environment, allow evolution of the mean phenotype to undergo a stochastic maximization of . Although the expected evolution maximizes the long-run growth rate of the population, for a genotype or phenotype the long-run growth rate is not a valid measure of fitness in a fluctuating environment. The haploid and quantitative character models both reveal that the expected relative fitness of a type is its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population.     

Multi-component gene network design as a survival strategy in diverse environments

Background

Gene-environment interactions are often mediated though gene networks in which gene expression products interact with other network components to dictate network activity levels, which in turn determines the fitness of the host cell in specific environments. Even though a gene network is the right context for studying gene-environment interactions, we have little understanding on how systematic genetic perturbations affects fitness in the context of a gene network.

Results

Here we examine the effect of combinatorial gene dosage alterations on gene network activity and cellular fitness. Using the galactose utilization pathway as a model network in diploid yeast, we reduce the copy number of four regulatory genes (GAL2, GAL3, GAL4, GAL80) from two to one, and measure the activity of the perturbed networks. We integrate these results with competitive fitness measurements made in six different rationally-designed environments containing different galactose concentrations representing the natural induction spectrum of the galactose network. In the lowest galactose environment, we find a nonlinear relationship between gene expression and fitness while high galactose environments lead to a linear relationship between the two with a saturation regime reached at a sufficiently high galactose concentration. We further uncover environment-specific relevance of the different network components for dictating the relationship between the network activity and organismal fitness, indicating that none of the network components are redundant.

Conclusions

These results provide experimental support to the hypothesis that dynamic changes in the environment throughout natural evolution is key to structuring natural gene networks in a multi-component fashion, which robustly provides protection against population extinction in different environments.

     

Stochastic dynamics of cancer initiation

Most human cancer types result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Once the first change (or changes) have arisen, tumorigenesis is initiated and the subsequent emergence of additional alterations drives progression to more aggressive and ultimately invasive phenotypes. Elucidation of the dynamics of cancer initiation is of importance for an understanding of tumor evolution and cancer incidence data. In this paper, we develop a novel mathematical framework to study the processes of cancer initiation. Cells at risk of accumulating oncogenic mutations are organized into small compartments of cells and proliferate according to a stochastic process. During each cell division, an (epi)genetic alteration may arise which leads to a random fitness change, drawn from a probability distribution. Cancer is initiated when a cell gains a fitness sufficiently high to escape from the homeostatic mechanisms of the cell compartment. To investigate cancer initiation during a human lifetime, a 'race' between this fitness process and the aging process of the patient is considered; the latter is modeled as a second stochastic Markov process in an aging dimension. This model allows us to investigate the dynamics of cancer initiation and its dependence on the mutational fitness distribution. Our framework also provides a methodology to assess the effects of different life expectancy distributions on lifetime cancer incidence. We apply this methodology to colorectal tumorigenesis while considering life expectancy data of the US population to inform the dynamics of the aging process. We study how the probability of cancer initiation prior to death, the time until cancer initiation, and the mutational profile of the cancer-initiating cell depends on the shape of the mutational fitness distribution and life expectancy of the population.     

From stochastic environments to life histories and back

Environmental stochasticity is known to play an important role in life-history evolution, but most general theory assumes a constant environment. In this paper, we examine life-history evolution in a variable environment, by decomposing average individual fitness (measured by the long-run stochastic growth rate) into contributions from average vital rates and their temporal variation. We examine how generation time, demographic dispersion (measured by the dispersion of reproductive events across the lifespan), demographic resilience (measured by damping time), within-year variances in vital rates, within-year correlations between vital rates and between-year correlations in vital rates combine to determine average individual fitness of stylized life histories. In a fluctuating environment, we show that there is often a range of cohort generation times at which the fitness is at a maximum. Thus, we expect ‘optimal’ phenotypes in fluctuating environments to differ from optimal phenotypes in constant environments. We show that stochastic growth rates are strongly affected by demographic dispersion, even when deterministic growth rates are not, and that demographic dispersion also determines the response of life-history-specific average fitness to within- and between-year correlations. Serial correlations can have a strong effect on fitness, and, depending on the structure of the life history, may act to increase or decrease fitness. The approach we outline takes a useful first step in developing general life-history theory for non-constant environments.     

Coexistence of trichome variation in a natural plant population: a combined study using ecological and candidate gene approaches

The coexistence of distinct phenotypes within populations has long been investigated in evolutionary ecology. Recent studies have identified the genetic basis of distinct phenotypes, but it is poorly understood how the variation in candidate loci is maintained in natural environments. In this study, we examined fitness consequences and genetic basis of variation in trichome production in a natural population of Arabidopsis halleri subsp. gemmifera. Half of the individuals in the study population produced trichomes while the other half were glabrous, and the leaf beetle Phaedon brassicae imposed intensive damage to both phenotypes. The fitness of hairy and glabrous plants showed no significant differences in the field during two years. A similar result was obtained when sibling hairy and glabrous plants were transplanted at the same field site, whereas a fitness cost of trichome production was detected under a weak herbivory condition. Thus, equivalent fitness of hairy and glabrous plants under natural herbivory allows their coexistence in the contemporary population. The pattern of polymorphism of the candidate trichome gene GLABROUS1 (GL1) showed no evidence of long-term maintenance of trichome variation within the population. Although balancing selection under fluctuating biotic environments is often proposed to explain the maintenance of defense variation, the lack of clear evidence of balancing selection in the study population suggests that other factors such as gene flow and neutral process may have played relatively large roles in shaping trichome variation at least for the single population level.