Microbiota as Predictor of Mortality in Allogeneic Hematopoietic-Cell Transplantation.

BACKGROUND: Relationships between microbiota composition and clinical outcomes after allogeneic hematopoietic-cell transplantation have been described in single-center studies. Geographic variations in the composition of human microbial communities and differences in clinical practices across institutions raise the question of whether these associations are generalizable. METHODS: The microbiota composition of fecal samples obtained from patients who were undergoing allogeneic hematopoietic-cell transplantation at four centers was profiled by means of 16S ribosomal RNA gene sequencing. In an observational study, we examined associations between microbiota diversity and mortality using Cox proportional-hazards analysis. For stratification of the cohorts into higher- and lower-diversity groups, the median diversity value that was observed at the study center in New York was used. In the analysis of independent cohorts, the New York center was cohort 1, and three centers in Germany, Japan, and North Carolina composed cohort 2. Cohort 1 and subgroups within it were analyzed for additional outcomes, including transplantation-related death. RESULTS: We profiled 8767 fecal samples obtained from 1362 patients undergoing allogeneic hematopoietic-cell transplantation at the four centers. We observed patterns of microbiota disruption characterized by loss of diversity and domination by single taxa. Higher diversity of intestinal microbiota was associated with a lower risk of death in independent cohorts (cohort 1: 104 deaths among 354 patients in the higher-diversity group vs. 136 deaths among 350 patients in the lower-diversity group; adjusted hazard ratio, 0.71; 95% confidence interval [CI], 0.55 to 0.92; cohort 2: 18 deaths among 87 patients in the higher-diversity group vs. 35 deaths among 92 patients in the lower-diversity group; adjusted hazard ratio, 0.49; 95% CI, 0.27 to 0.90). Subgroup analyses identified an association between lower intestinal diversity and higher risks of transplantation-related death and death attributable to graft-versus-host disease. Baseline samples obtained before transplantation already showed evidence of microbiome disruption, and lower diversity before transplantation was associated with poor survival. CONCLUSIONS: Patterns of microbiota disruption during allogeneic hematopoietic-cell transplantation were similar across transplantation centers and geographic locations; patterns were characterized by loss of diversity and domination by single taxa. Higher diversity of intestinal microbiota at the time of neutrophil engraftment was associated with lower mortality. (Funded by the National Cancer Institute and others.).

Abnormal morphology biases hematocrit distribution in tumor vasculature and contributes to heterogeneity in tissue oxygenation.

Oxygen heterogeneity in solid tumors is recognized as a limiting factor for therapeutic efficacy. This heterogeneity arises from the abnormal vascular structure of the tumor, but the precise mechanisms linking abnormal structure and compromised oxygen transport are only partially understood. In this paper, we investigate the role that red blood cell (RBC) transport plays in establishing oxygen heterogeneity in tumor tissue. We focus on heterogeneity driven by network effects, which are challenging to observe experimentally due to the reduced fields of view typically considered. Motivated by our findings of abnormal vascular patterns linked to deviations from current RBC transport theory, we calculated average vessel lengths [Formula: see text] and diameters [Formula: see text] from tumor allografts of three cancer cell lines and observed a substantial reduction in the ratio [Formula: see text] compared to physiological conditions. Mathematical modeling reveals that small values of the ratio λ (i.e., [Formula: see text]) can bias hematocrit distribution in tumor vascular networks and drive heterogeneous oxygenation of tumor tissue. Finally, we show an increase in the value of λ in tumor vascular networks following treatment with the antiangiogenic cancer agent DC101. Based on our findings, we propose λ as an effective way of monitoring the efficacy of antiangiogenic agents and as a proxy measure of perfusion and oxygenation in tumor tissue undergoing antiangiogenic treatment.

A multiscale model of complex endothelial cell dynamics in early angiogenesis.

We introduce a hybrid two-dimensional multiscale model of angiogenesis, the process by which endothelial cells (ECs) migrate from a pre-existing vascular bed in response to local environmental cues and cell-cell interactions, to create a new vascular network. Recent experimental studies have highlighted a central role of cell rearrangements in the formation of angiogenic networks. Our model accounts for this phenomenon via the heterogeneous response of ECs to their microenvironment. These cell rearrangements, in turn, dynamically remodel the local environment. The model reproduces characteristic features of angiogenic sprouting that include branching, chemotactic sensitivity, the brush border effect, and cell mixing. These properties, rather than being hardwired into the model, emerge naturally from the gene expression patterns of individual cells. After calibrating and validating our model against experimental data, we use it to predict how the structure of the vascular network changes as the baseline gene expression levels of the VEGF-Delta-Notch pathway, and the composition of the extracellular environment, vary. In order to investigate the impact of cell rearrangements on the vascular network structure, we introduce the mixing measure, a scalar metric that quantifies cell mixing as the vascular network grows. We calculate the mixing measure for the simulated vascular networks generated by ECs of different lineages (wild type cells and mutant cells with impaired expression of a specific receptor). Our results show that the time evolution of the mixing measure is directly correlated to the generic features of the vascular branching pattern, thus, supporting the hypothesis that cell rearrangements play an essential role in sprouting angiogenesis. Furthermore, we predict that lower cell rearrangement leads to an imbalance between branching and sprout elongation. Since the computation of this statistic requires only individual cell trajectories, it can be computed for networks generated in biological experiments, making it a potential biomarker for pathological angiogenesis.

Bivalent chromatin as a therapeutic target in cancer: An in silico predictive approach for combining epigenetic drugs.

Tumour cell heterogeneity is a major barrier for efficient design of targeted anti-cancer therapies. A diverse distribution of phenotypically distinct tumour-cell subpopulations prior to drug treatment predisposes to non-uniform responses, leading to the elimination of sensitive cancer cells whilst leaving resistant subpopulations unharmed. Few strategies have been proposed for quantifying the variability associated to individual cancer-cell heterogeneity and minimizing its undesirable impact on clinical outcomes. Here, we report a computational approach that allows the rational design of combinatorial therapies involving epigenetic drugs against chromatin modifiers. We have formulated a stochastic model of a bivalent transcription factor that allows us to characterise three different qualitative behaviours, namely: bistable, high- and low-gene expression. Comparison between analytical results and experimental data determined that the so-called bistable and high-gene expression behaviours can be identified with undifferentiated and differentiated cell types, respectively. Since undifferentiated cells with an aberrant self-renewing potential might exhibit a cancer/metastasis-initiating phenotype, we analysed the efficiency of combining epigenetic drugs against the background of heterogeneity within the bistable sub-ensemble. Whereas single-targeted approaches mostly failed to circumvent the therapeutic problems represented by tumour heterogeneity, combinatorial strategies fared much better. Specifically, the more successful combinations were predicted to involve modulators of the histone H3K4 and H3K27 demethylases KDM5 and KDM6A/UTX. Those strategies involving the H3K4 and H3K27 methyltransferases MLL2 and EZH2, however, were predicted to be less effective. Our theoretical framework provides a coherent basis for the development of an in silico platform capable of identifying the epigenetic drugs combinations best-suited to therapeutically manage non-uniform responses of heterogenous cancer cell populations.

Microvascular alterations in children with optic disk drusen evaluated by optical coherence tomography angiography.

PURPOSE: To asses changes in vessel density (VD) in children with optic disk drusen (ODD) using swept source optical coherence tomography angiography (OCTA). METHODS: Cross-sectional study of 27 eyes with ODD compared with age-matched controls. Peripapillary and macular VD were measured in the superficial retinal capillary plexus (SCP), deep capillary plexus (DCP), and choriocapillaris (CC). The correlation between VD changes with alterations in retinal nerve fiber layer (RNFL), ganglion cell layer (GCL), and visual field (VF) was analyzed. RESULTS: Mean participant age was 12.5 ± 3.3 years (range, 7-18 years); 63% was females. In the patients vs. controls, median central peripapillary VD was 52.9% vs. 50.6% (p = 0.63) for SCP; 48.1% vs. 53.8% (p = 0.017) for DCP; and 17.0% vs. 28.2% (p = 0.0037) for CC, respectively. VD in the superior and nasal CC layers was significantly lower in the patients (36.3% vs. 56.2%; p < 0.001) and (60.4% vs. 70.3%, p < 0.001), respectively. No significant differences were observed for VD in the macular region. The RNFL was thinner in eyes with superficial drusen versus controls (87 vs. 111 µm; p < 0.001). No significant differences between were observed in GCL thickness (p = 0.13). Nasal SCP and nasal RNFL VD were moderately correlated (r = 0.54, p < 0.01), while mean VF deviation was strongly correlated with median SCP VD in patients with superficial drusen (r = 0.9, p = 0.03). CONCLUSION: Impaired VD was observed in the peripapillary nasal CC in patients with ODD; this impairment was associated with a decreased RNFL thickness. Nasal SCP VD and RNFL thickness were moderately correlated in patients with ODD.

A multiscale model of epigenetic heterogeneity-driven cell fate decision-making.

The inherent capacity of somatic cells to switch their phenotypic status in response to damage stimuli in vivo might have a pivotal role in ageing and cancer. However, how the entry-exit mechanisms of phenotype reprogramming are established remains poorly understood. In an attempt to elucidate such mechanisms, we herein introduce a stochastic model of combined epigenetic regulation (ER)-gene regulatory network (GRN) to study the plastic phenotypic behaviours driven by ER heterogeneity. To deal with such complex system, we additionally formulate a multiscale asymptotic method for stochastic model reduction, from which we derive an efficient hybrid simulation scheme. Our analysis of the coupled system reveals a regime of tristability in which pluripotent stem-like and differentiated steady-states coexist with a third indecisive state, with ER driving transitions between these states. Crucially, ER heterogeneity of differentiation genes is for the most part responsible for conferring abnormal robustness to pluripotent stem-like states. We formulate epigenetic heterogeneity-based strategies capable of unlocking and facilitating the transit from differentiation-refractory (stem-like) to differentiation-primed epistates. The application of the hybrid numerical method validates the likelihood of such switching involving solely kinetic changes in epigenetic factors. Our results suggest that epigenetic heterogeneity regulates the mechanisms and kinetics of phenotypic robustness of cell fate reprogramming. The occurrence of tunable switches capable of modifying the nature of cell fate reprogramming might pave the way for new therapeutic strategies to regulate reparative reprogramming in ageing and cancer.

Viral replication modes in single-peak fitness landscapes: A dynamical systems analysis.

Positive-sense, single-stranded RNA viruses are important pathogens infecting almost all types of organisms. Experimental evidence from distributions of mutations and from viral RNA amplification suggest that these pathogens may follow different RNA replication modes, ranging from the stamping machine replication (SMR) to the geometric replication (GR) mode. Although previous theoretical work has focused on the evolutionary dynamics of RNA viruses amplifying their genomes with different strategies, little is known in terms of the bifurcations and transitions involving the so-called error threshold (mutation-induced dominance of mutants) and lethal mutagenesis (extinction of all sequences due to mutation accumulation and demographic stochasticity). Here we analyze a dynamical system describing the intracellular amplification of viral RNA genomes evolving on a single-peak fitness landscape focusing on three cases considering neutral, deleterious, and lethal mutants. We analytically derive the critical mutation rates causing lethal mutagenesis and error threshold, governed by transcritical bifurcations that depend on parameters α (parameter introducing the mode of replication), replicative fitness of mutants (k1), and on the spontaneous degradation rates of the sequences (ϵ). Our results relate the error catastrophe with lethal mutagenesis in a model with continuous populations of viral genomes. The former case involves dominance of the mutant sequences, while the latter, a deterministic extinction of the viral RNAs during replication due to increased mutation. For the lethal case the critical mutation rate involving lethal mutagenesis is µc=1-ɛ/α. Here, the SMR involves lower critical mutation rates, being the system more robust to lethal mutagenesis replicating closer to the GR mode. This result is also found for the neutral and deleterious cases, but for these later cases lethal mutagenesis can shift to the error threshold once the replication mode surpasses a threshold given by α=ϵ/k1.

Epigenetic regulation of cell fate reprogramming in aging and disease: A predictive computational model.

Understanding the control of epigenetic regulation is key to explain and modify the aging process. Because histone-modifying enzymes are sensitive to shifts in availability of cofactors (e.g. metabolites), cellular epigenetic states may be tied to changing conditions associated with cofactor variability. The aim of this study is to analyse the relationships between cofactor fluctuations, epigenetic landscapes, and cell state transitions. Using Approximate Bayesian Computation, we generate an ensemble of epigenetic regulation (ER) systems whose heterogeneity reflects variability in cofactor pools used by histone modifiers. The heterogeneity of epigenetic metabolites, which operates as regulator of the kinetic parameters promoting/preventing histone modifications, stochastically drives phenotypic variability. The ensemble of ER configurations reveals the occurrence of distinct epi-states within the ensemble. Whereas resilient states maintain large epigenetic barriers refractory to reprogramming cellular identity, plastic states lower these barriers, and increase the sensitivity to reprogramming. Moreover, fine-tuning of cofactor levels redirects plastic epigenetic states to re-enter epigenetic resilience, and vice versa. Our ensemble model agrees with a model of metabolism-responsive loss of epigenetic resilience as a cellular aging mechanism. Our findings support the notion that cellular aging, and its reversal, might result from stochastic translation of metabolic inputs into resilient/plastic cell states via ER systems.

Age Structure Can Account for Delayed Logistic Proliferation of Scratch Assays.

Scratch assays are in vitro methods for studying cell migration. In these experiments, a scratch is made on a cell monolayer and recolonisation of the scratched region is imaged to quantify cell migration rates. Typically, scratch assays are modelled by reaction diffusion equations depicting cell migration by Fickian diffusion and proliferation by a logistic term. In a recent paper (Jin et al. in Bull Math Biol 79(5):1028-1050, 2017), the authors observed experimentally that during the early stage of the recolonisation process, there is a disturbance phase where proliferation is not logistic, and this is followed by a growth phase where proliferation appears to be logistic. The authors did not identify the precise mechanism that causes the disturbance phase but showed that ignoring it can lead to incorrect parameter estimates. The aim of this work is to show that a nonlinear age-structured population model can account for the two phases of proliferation in scratch assays. The model consists of an age-structured cell cycle model of a cell population, coupled with an ordinary differential equation describing the resource concentration dynamics in the substrate. The model assumes a resource-dependent cell cycle threshold age, above which cells are able to proliferate. By studying the dynamics of the full system in terms of the subpopulations of cells that can proliferate and the ones that can not, we are able to find conditions under which the model captures the two-phase behaviour. Through numerical simulations, we are able to show that the interplay between the resource concentration in the substrate and the cell subpopulations dynamics can explain the biphasic dynamics.

Minimum Action Path Theory Reveals the Details of Stochastic Transitions Out of Oscillatory States.

Cell state determination is the outcome of intrinsically stochastic biochemical reactions. Transitions between such states are studied as noise-driven escape problems in the chemical species space. Escape can occur via multiple possible multidimensional paths, with probabilities depending nonlocally on the noise. Here we characterize the escape from an oscillatory biochemical state by minimizing the Freidlin-Wentzell action, deriving from it the stochastic spiral exit path from the limit cycle. We also use the minimized action to infer the escape time probability density function.

3D hybrid modelling of vascular network formation.

We develop an off-lattice, agent-based model to describe vasculogenesis, the de novo formation of blood vessels from endothelial progenitor cells during development. The endothelial cells that comprise our vessel network are viewed as linearly elastic spheres that move in response to the forces they experience. We distinguish two types of endothelial cells: vessel elements are contained within the network and tip cells are located at the ends of vessels. Tip cells move in response to mechanical forces caused by interactions with neighbouring vessel elements and the local tissue environment, chemotactic forces and a persistence force which accounts for their tendency to continue moving in the same direction. Vessel elements are subject to similar mechanical forces but are insensitive to chemotaxis. An angular persistence force representing interactions with the local tissue is introduced to stabilise buckling instabilities caused by cell proliferation. Only vessel elements proliferate, at rates which depend on their degree of stretch: elongated elements have increased rates of proliferation, and compressed elements have reduced rates. Following division, the fate of the new cell depends on the local mechanical environment: the probability of forming a new sprout is increased if the parent vessel is highly compressed and the probability of being incorporated into the parent vessel increased if the parent is stretched. Simulation results reveal that our hybrid model can reproduce the key qualitative features of vasculogenesis. Extensive parameter sensitivity analyses show that significant changes in network size and morphology are induced by varying the chemotactic sensitivity of tip cells, and the sensitivities of the proliferation rate and the sprouting probability to mechanical stretch. Varying the chemotactic sensitivity directly influences the directionality of the networks. The degree of branching, and thereby the density of the networks, is influenced by the sprouting probability. Glyphs that simultaneously depict several network properties are introduced to show how these and other network quantities change over time and also as model parameters vary. We also show how equivalent glyphs constructed from in vivo data could be used to discriminate between normal and tumour vasculature and, in the longer term, for model validation. We conclude that our biomechanical hybrid model can generate vascular networks that are qualitatively similar to those generated from in vitro and in vivo experiments.

Evolutionary escape on complex genotype-phenotype networks.

We study the problem of evolutionary escape that is the process whereby a population under sudden changes in the selective pressures acting upon it try to evade extinction by evolving from previously well-adapted phenotypes to those that are favoured by the new selective pressure. We perform a comparative analysis between results obtained by modelling genotype space as a regular hypercube (H-graphs), which is the scenario considered in previous work on the subject, to those corresponding to a complex genotype-phenotype network (B-graphs). In order to analyse the properties of the escape process on both these graphs, we apply a general theory based on multi-type branching processes to compute the evolutionary dynamics and probability of escape. We show that the distribution of distances between phenotypes in B-graphs exhibits a much larger degree of heterogeneity than in H-graphs. This property, one of the main structural differences between both types of graphs, causes heterogeneous behaviour in all results associated to the escape problem. We further show that, due to the heterogeneity characterising escape on B-graphs, escape probability can be underestimated by assuming a regular hypercube genotype network, even if we compare phenotypes at the same distance in H-graphs. Similarly, it appears that the complex structure of B-graphs slows down the rate of escape.

Stochastic multi-scale models of competition within heterogeneous cellular populations: Simulation methods and mean-field analysis.

We propose a modelling framework to analyse the stochastic behaviour of heterogeneous, multi-scale cellular populations. We illustrate our methodology with a particular example in which we study a population with an oxygen-regulated proliferation rate. Our formulation is based on an age-dependent stochastic process. Cells within the population are characterised by their age (i.e. time elapsed since they were born). The age-dependent (oxygen-regulated) birth rate is given by a stochastic model of oxygen-dependent cell cycle progression. Once the birth rate is determined, we formulate an age-dependent birth-and-death process, which dictates the time evolution of the cell population. The population is under a feedback loop which controls its steady state size (carrying capacity): cells consume oxygen which in turn fuels cell proliferation. We show that our stochastic model of cell cycle progression allows for heterogeneity within the cell population induced by stochastic effects. Such heterogeneous behaviour is reflected in variations in the proliferation rate. Within this set-up, we have established three main results. First, we have shown that the age to the G1/S transition, which essentially determines the birth rate, exhibits a remarkably simple scaling behaviour. Besides the fact that this simple behaviour emerges from a rather complex model, this allows for a huge simplification of our numerical methodology. A further result is the observation that heterogeneous populations undergo an internal process of quasi-neutral competition. Finally, we investigated the effects of cell-cycle-phase dependent therapies (such as radiation therapy) on heterogeneous populations. In particular, we have studied the case in which the population contains a quiescent sub-population. Our mean-field analysis and numerical simulations confirm that, if the survival fraction of the therapy is too high, rescue of the quiescent population occurs. This gives rise to emergence of resistance to therapy since the rescued population is less sensitive to therapy.

Surviving evolutionary escape on complex genotype-phenotype networks.

We study the problem of evolutionary escape and survival of cell populations with a genotype-phenotype structure. We refer to evolutionary escape as the process where a cell of a given ill-adapted population to reach a well-adapted phenotype. Similarly, survival refers to the dynamics of the population once the escape phenotype has been reached. The aim of this paper is to analyse the influence of topological properties associated to robustness and evolvability on the probability of escape and on the probability of survival. In order to explore these issues, we formulate a population dynamics model, consisting of a multi-type time-continuous branching process, where types are associated to genotypes and their birth and death probabilities depend on the associated phenotype (non-escape or escape). We exploit the separation of time scales introduced by the the difference in reproductive ratios between the ill-adapted phenotypes and the escape phenotype. Two dynamical regimes emerge: a fast-decaying regime associated to the escape process itself, and a slow regime which corresponds to the survival dynamics of the population once the escape phenotype has been reached. We exploit this separation of time scales to analyse the topological factors which determine escape and survival probabilities. We show that, while the escape probability depends on the degree of escape phenotype, the probability of survival is essentially determined by its robustness, measured in terms of a weighted clustering coefficient.

Stochastic modelling of the eradication of the HIV-1 infection by stimulation of latently infected cells in patients under highly active anti-retroviral therapy.

HIV-1 infected patients are effectively treated with highly active anti-retroviral therapy (HAART). Whilst HAART is successful in keeping the disease at bay with average levels of viral load well below the detection threshold of standard clinical assays, it fails to completely eradicate the infection, which persists due to the emergence of a latent reservoir with a half-life time of years and is immune to HAART. This implies that life-long administration of HAART is, at the moment, necessary for HIV-1-infected patients, which is prone to drug resistance and cumulative side effects as well as imposing a considerable financial burden on developing countries, those more afflicted by HIV, and public health systems. The development of therapies which specifically aim at the removal of this latent reservoir has become a focus of much research. A proposal for such therapy consists of elevating the rate of activation of the latently infected cells: by transferring cells from the latently infected reservoir to the active infected compartment, more cells are exposed to the anti-retroviral drugs thus increasing their effectiveness. In this paper, we present a stochastic model of the dynamics of the HIV-1 infection and study the effect of the rate of latently infected cell activation on the average extinction time of the infection. By analysing the model by means of an asymptotic approximation using the semi-classical quasi steady state approximation (QSS), we ascertain that this therapy reduces the average life-time of the infection by many orders of magnitudes. We test the accuracy of our asymptotic results by means of direct simulation of the stochastic process using a hybrid multi-scale Monte Carlo scheme.

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

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, PS(t), has a fat-tail behaviour (PS(t) ∼ 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.

Stochastic modelling of viral blips in HIV-1-infected patients: effects of inhomogeneous density fluctuations.

We propose a stochastic model of HIV-1 infection dynamics under HAART in order to analyse the origin and dynamics of the so-called viral blips, i.e. episodes of transient viremia that occur in the phase of where the disease remains in a latent state during which the viral load raises above the detection limit of standard clinical assays. Based on prior work in the subject, we consider an infection model in which latently infected cell compartment sustains a residual (latent) infection over long periods of time. Unlike previous models, we include the effects of inhomogeneities in cell and virus concentration in the blood stream. We further consider the effect of burst virion production. By comparing with the experimental results obtained during a study in which intensive sampling was carried out on HIV-1-infected patients undergoing HAART over a long period of time, we conclude that our model supports the hypothesis that viral blips are consistent with random fluctuations around the average viral load. We further observe that agreement between our simulation results and the blip statistics obtained in the aforementioned study improves when burst virion production is considered. We also study the effect of sample manipulation artifacts on the results produced by our model, in particular, that of the post-extraction handling time, i.e. the time elapsed between sample extraction and actual test. Our results support the notion that the statistics of viral blips can be critically affected by such artifacts.

The effects of intrinsic noise on the behaviour of bistable cell regulatory systems under quasi-steady state conditions.

We analyse the effect of intrinsic fluctuations on the properties of bistable stochastic systems with time scale separation operating under quasi-steady state conditions. We first formulate a stochastic generalisation of the quasi-steady state approximation based on the semi-classical approximation of the partial differential equation for the generating function associated with the chemical master equation. Such approximation proceeds by optimising an action functional whose associated set of Euler-Lagrange (Hamilton) equations provides the most likely fluctuation path. We show that, under appropriate conditions granting time scale separation, the Hamiltonian can be re-scaled so that the set of Hamilton equations splits up into slow and fast variables, whereby the quasi-steady state approximation can be applied. We analyse two particular examples of systems whose mean-field limit has been shown to exhibit bi-stability: an enzyme-catalysed system of two mutually inhibitory proteins and a gene regulatory circuit with self-activation. Our theory establishes that the number of molecules of the conserved species is order parameters whose variation regulates bistable behaviour in the associated systems beyond the predictions of the mean-field theory. This prediction is fully confirmed by direct numerical simulations using the stochastic simulation algorithm. This result allows us to propose strategies whereby, by varying the number of molecules of the three conserved chemical species, cell properties associated to bistable behaviour (phenotype, cell-cycle status, etc.) can be controlled.

The topology of robustness and evolvability in evolutionary systems with genotype-phenotype map.

In this paper we formulate a topological definition of the concepts of robustness and evolvability. We start our investigation by formulating a multiscale model of the evolutionary dynamics of a population of cells. Our cells are characterised by a genotype-phenotype map: their chances of survival under selective pressure are determined by their phenotypes, whereas the latter are determined their genotypes. According to our multiscale dynamics, the population dynamics generates the evolution of a genotype-phenotype network. Our representation of the genotype-phenotype network is similar to previously described ones, but has a novel element, namely, our network contains two types of nodes: genotype and phenotype nodes. This network representation allows us to characterise robustness and evolvability in terms of its topological properties: phenotypic robustness by means of the clustering coefficient of the phenotype nodes, and evolvability as the emergence of giant connected component which allows navigation between phenotypes. This topological definition of evolvability allows us to characterise the so-called robustness of evolvability, which is defined in terms of the robustness against attack (i.e. edge removal) of the giant connected component. An investigation of the factors that affect the robustness of evolvability shows that phenotypic robustness and the cryptic genetic variation are key to the integrity of the ability to innovate. These results fit within the framework of a number of models which point out that robustness favours rather than hindering evolvability. We further show that the corresponding phenotype network, defined as the one-component projection of the whole genotype-phenotype network, exhibits the small-world phenomenon, which implies that in this type of evolutionary system the rate of adaptability is enhanced.