We show how field- and information theory can be used to quantify the relationship between genotype and phenotype in cases where phenotype is a continuous variable. Given a sample population of phenotype measurements, from various known genotypes, we show how the ordering of phenotype data can lead to quantification of the effect of genotype. This method does not assume that the data has a Gaussian distribution, it is particularly effective at extracting weak and unusual dependencies of genotype on phenotype. However, in cases where data has a special form, (eg Gaussian), we observe that the effective phenotype field has a special form. We use asymptotic analysis to solve both the forward and reverse formulations of the problem. We show how p-values can be calculated so that the significance of correlation between phenotype and genotype can be quantified. This provides a significant generalisation of the traditional methods used in genome-wide association studies GWAS. We derive a field-strength which can be used to deduce how the correlations between genotype and phenotype, and their impact on the distribution of phenotypes.

Invasive plant species pose a significant threat to biodiversity and the economy, yet their management is often resource-intensive and expensive, and further research is required to make control measures more efficient. Evidence suggests that roads can have an important effect on the spread of invasive plant species, although little is known about the underlying mechanisms at play. We have developed a novel mathematical model to analyse the impact of roads on the propagation of invasive plants. The integro-difference equation model is formulated for stage-structured population and incorporates a road sub-domain in the spatial domain. The results of our study reveal, that, depending on the definition of the growth function in the model, there are three distinct types of behaviour in front of the road. Roads can act as barriers to invasion, lead to a formation of a beachhead in front of the road, or act as corridors allowing the invasive species to invade the domain in front of the road. Analytical and computational findings on how roads can impact the spread of invasive species show that a small change in conditions of the environment favouring the invasive species can change the case for the road, allowing the invasive species to invade the domain in front of the road where it previously could not spread.

Reputation is one of key mechanisms to maintain human cooperation, but its analysis gets complicated if we consider the possibility that reputation does not reach consensus because of erroneous assessment. The difficulty is alleviated if we assume that reputation and cooperation do not take binary values but have continuous spectra so that disagreement over reputation can be analysed in a perturbative way. In this work, we carry out the analysis by expanding the dynamics of reputation to the second order of perturbation under the assumption that everyone initially cooperates with good reputation. The second-order theory clarifies the difference between Image Scoring and Simple Standing in that punishment for defection against a well-reputed player should be regarded as good for maintaining cooperation. Moreover, comparison among the leading eight shows that the stabilizing effect of justified punishment weakens if cooperation between two ill-reputed players is regarded as bad. Our analysis thus explains how Simple Standing achieves a high level of stability by permitting justified punishment and also by disregarding irrelevant information in assessing cooperation. This observation suggests which factors affect the stability of a social norm when reputation can be perturbed by noise.

The linear decrease in the frequency of blue whale songs around the world is, to date, an unexplained phenomenon. We show it can be reproduced by a mathematical model considering two antagonistic behavioral trends: first, a bias towards conformity in the song, and second, a tendency to try and sing lower than the other whales. We check the robustness of our model by considering some more complex premises. First, different hierarchical relations between the singers are explored, adapting methods used in the flocking motion studies. Then a population-dependant simulation shows that even considering the gradual addition of new whales, the evolution is still globally linear. Finally, we show that intra-annual variations surging from different causes can be naturally incorporated into the model. We then conclude that, unlike other explanations, a cultural hypothesis seems compatible with the observed linearity of the blue whales's songs frequency shift.

Pancreatic Ductal Adenocarcinoma (PDAC) is widely known for its poor prognosis because it is often diagnosed when the cancer is in a later stage. We built a Boolean model to analyze the microenvironment of pancreatic cancer in order to better understand the interplay between pancreatic cancer, stellate cells, and their signaling cytokines. Specifically, we have used our model to study the impact of inducing four common mutations: KRAS, TP53, SMAD4, and CDKN2A. After implementing the various mutation combinations, we used our stochastic simulator to derive aggressiveness scores based on simulated attractor probabilities and long-term trajectory approximations. These aggression scores were then corroborated with clinical data. Moreover, we found sets of control targets that are effective among common mutations. These control sets contain nodes within both the pancreatic cancer cell and the pancreatic stellate cell, including PIP3, RAF, PIK3 and BAX in pancreatic cancer cell as well as ERK and PIK3 in the pancreatic stellate cell. Many of these nodes were found to be differentially expressed among pancreatic cancer patients in the TCGA database. Furthermore, literature suggests that many of these nodes can be targeted by drugs currently in circulation. The results herein help provide a proof of concept in the path towards personalized medicine through a means of mathematical systems biology. All data and code used for running simulations, statistical analysis, and plotting is available on a GitHub repository athttps://github.com/drplaugher/PCC_Mutations.

Multi-strain pathogens such as Group A Streptococcus, Streptococcus pneumoniae, and Staphylococcus aureus cause millions of infections each year with a substantial health burden. Control of multi-strain pathogens can be complicated by the high strain diversity often observed in endemic settings. It is not well understood how high strain diversity is maintained in populations, given that they compete with each other both directly (within an individual host) and indirectly (via host immunity). Previous modelling studies have investigated how indirect competition affects the prevalence and diversity of strains. However, these studies often make simplifying assumptions about the direct competition that occurs within hosts. Currently, little data is available to validate these assumptions, hence there is a need to clarify how sensitive model outputs are to these assumptions. In this study, we compare the dynamics of multi-strain pathogens under different assumptions about direct competition between strains using an agent-based model. We find that the assumptions made about direct competition can affect the epidemiological dynamics, particularly when there is no long-term immunity following infections and a low rate of importation of non-circulating strains. Our results suggest that while direct and indirect competition can each decrease strain diversity when they act in isolation, they may increase strain diversity when they act together. This finding highlights the importance of examining sensitivity to assumptions about strain competition. In particular, omitting consideration of direct competition can lead to inaccurate estimates of the likely effectiveness of control strategies as changes in strain diversity shift the level of direct strain competition.

In recent years, new research programmes have been initiated to understand the role of gut bacteria in health and disease, enabled in large part by the emergence of high-throughput sequencing. As new genomic and other data emerge it will become important to explain observations in terms of underlying population mechanisms; for instance, it is of interest to understand how resident bacteria interact with their hosts and pathogens, and how they play a protective role. Connecting underlying processes with observed patterns is aided by the development of mathematical models. Here, we develop a spatial model of microbial populations in the gastrointestinal tract to explore conditions under which inflammation-causing bacteria can invade the gut and under which such pathogens become persistent. We find that pathogens invade both small and large intestine from even a relatively small inoculum size but are usually eliminated by the host response. When the immune response is weak, the pathogen is able to persist for a long period. Spatial structure affects these dynamics by creating moving refugia which facilitate bouts of pathogen resurgence and inflammation in persistent infections. Space also plays a role in repopulation by commensals after infection. We further find that the rate of decay of inflammation has a stronger effect on outcomes than the initiation of inflammation or other parameters. Finally, we explore the impact of partially inflammation-resistant commensals on these dynamics.

The coalescent model represents how individuals sampled from a population may have originated from a last common ancestor. The bounded coalescent model is obtained by conditioning the coalescent model such that the last common ancestor must have existed after a certain date. This conditioned model arises in a variety of applications, such as speciation, horizontal gene transfer or transmission analysis, and yet the bounded coalescent model has not been previously analysed in detail. Here we describe a new algorithm to simulate from this model directly, without resorting to rejection sampling. We show that this direct simulation algorithm is more computationally efficient than the rejection sampling approach. We also show how to calculate the probability of the last common ancestor occurring after a given date, which is required to compute the probability density of realisations under the bounded coalescent model. Our results are applicable in both the isochronous (when all samples have the same date) and heterochronous (where samples can have different dates) settings. We explore the effect of setting a bound on the date of the last common ancestor, and show that it affects a number of properties of the resulting phylogenies. All our methods are implemented in a new R package called BoundedCoalescent which is freely available online.

Seasonal variations in environmental conditions lead to changing infectious disease epidemic risks at different times of year. The probability that early cases initiate a major epidemic depends on the season in which the pathogen enters the population. The instantaneous epidemic risk (IER) can be tracked. This quantity is straightforward to calculate, and corresponds to the probability of a major epidemic starting from a single case introduced at time t=t0, assuming that environmental conditions remain identical from that time onwards (i.e. for all t≥t0). However, the threat when a pathogen enters the population in fact depends on changes in environmental conditions occurring within the timescale of the initial phase of the outbreak. For that reason, we compare the IER with a different metric: the case epidemic risk (CER). The CER corresponds to the probability of a major epidemic starting from a single case entering the population at time t=t0, accounting for changes in environmental conditions after that time. We show how the IER and CER can be calculated using different epidemiological models (the stochastic Susceptible-Infectious-Removed model and a stochastic host-vector model that is parameterised using temperature data for Miami) in which transmission parameter values vary temporally. While the IER is always easy to calculate numerically, the adaptable method we provide for calculating the CER for the host-vector model can also be applied easily and solved using widely available software tools. In line with previous research, we demonstrate that, if a pathogen is likely to either invade the population or fade out on a fast timescale compared to changes in environmental conditions, the IER closely matches the CER. However, if this is not the case, the IER and the CER can be significantly different, and so the CER should be used. This demonstrates the need to consider future changes in environmental conditions carefully when assessing the risk posed by emerging pathogens.

Gamma oscillations are a prominent feature of various neural systems, including the CA3 subfield of the hippocampus. In CA3, in vitro carbachol application induces ∼40 Hz gamma oscillations in the network of glutamatergic excitatory pyramidal neurons (PNs) and local GABAergic inhibitory neurons (INs). Activation of NMDA receptors within CA3 leads to an increase in the frequency of carbachol-induced oscillations to ∼60 Hz, a broadening of the distribution of individual oscillation cycle frequencies, and a decrease in the time lag between PN and IN spike bursts. In this work, we develop a biophysical integrate-and-fire model of the CA3 subfield, we show that the dynamics of our model are in concordance with physiological observations, and we provide computational support for the hypothesis that the 'E-I' mechanism is responsible for the emergence of ∼40 Hz gamma oscillations in the absence of NMDA activation. We then incorporate NMDA receptors into our CA3 model, and we show that our model exhibits the increase in gamma oscillation frequency, broadening of the cycle frequency distribution, and decrease in the time lag between PN and IN spike bursts observed experimentally. Remarkably, we find an inverse relationship in our model between the net NMDA current delivered to PNs and INs in an oscillation cycle and cycle frequency. Furthermore, we find a disparate effect of NMDA receptors on PNs versus INs - we show that NMDA receptors on INs tend to increase oscillation frequency, while NMDA receptors on PNs tend to slightly decrease or not affect oscillation frequency. We find that these observations can be explained if NMDA activity above a threshold level causes a shift in the mechanism underlying gamma oscillations; in the absence of NMDA receptors, the 'E-I' mechanism is primarily responsible for the generation of gamma oscillations (at 40 Hz), while when NMDA receptors are active, the mechanism of gamma oscillations shifts to the 'I-I' mechanism, and we argue that within the 'I-I' regime (which displays a higher baseline oscillation frequency of ∼60 Hz), slight changes in the level of NMDA activity are inversely related to cycle frequency.

Identifying mechanisms underlying neurons ability to process information including acquisition, storage, and retrieval plays an important role in the understanding of the different types of memory, pathogenesis of many neurological diseases affecting memory and therapeutic target discovery. However, the traditional understanding of the mechanisms of memory associated with the electrical signals having a unique combination of frequency and amplitude does not answer the question how the memories can survive for life-long periods of time, while exposed to synaptic noise. Recent evidence suggests that, apart from neuronal circuits, a diversity of the molecular memory (MM) carriers, are essential for memory performance. The molecular model of memory (MMM) is proposed, according to which each item of incoming information (the elementary memory item - eMI) is encoded by both circuitries, with the unique for a given MI electrical parameters, and also the MM carriers, unique by its molecular composition. While operating as the carriers of incoming information, the MMs, are functioning within the neuron plasma membrane. Inactive (latent) initially, during acquisition each of the eMIs is activated to become a virtual copy of some real fact or events bygone. This activation is accompanied by the considerable remodeling of the MM molecule associated with the resonance effect.

In this paper, we propose an easy to implement generalized linear models (GLM) methodology for estimating the basic reproduction number, R0, a major epidemic parameter for assessing the transmissibility of an infection. Our approach rests on well known qualitative properties of the classical SIR and SEIR systems for large populations. Moreover, we assume that information at the individual network level is not available. In inference we consider non homogeneous Poisson observation processes and mainly concentrate on epidemics that spread through a completely susceptible population. Further, we examine the performance of the estimator under various scenarios of relevance in practice, like partially observed data. We perform a detailed simulation study and illustrate our approach on Covid-19 Canadian data sets. Finally, we present extensions of our methodology and discuss its merits and practical limitations, in particular the challenges in estimating R0 when mitigation measures are applied.

Several studies have confirmed the existence of zero-determinant (ZD) strategies in repeated social dilemmas since Press and Dyson's ingenious discovery of ZD strategies in iterated prisoner's dilemmas. However, less research studies evolutionary performance of multiplayer ZD strategies, especially from a theoretical perspective. Here, we use a state-clustering method to theoretically analyze evolutionary dynamics of two representative ZD strategies: generous ZD strategies and extortionate ZD strategies. We consider two new settings for multiplayer ZD strategies: competitions with all ZD strategies and competitions with all memory-one strategies, apart from the competitions between these strategies and some classical ones. Moreover, we investigate the influence of the level of generosity and extortion on evolutionary dynamics of generous and extortionate ZD strategies, which was commonly ignored in previous studies. Theoretical results show that players with limited generosity are at an advantageous place and extortioners extorting more severely hold their ground more readily. Our results may provide new insights into better understanding evolutionary dynamics of ZD strategies in repeated multiplayer games.

The retinal vasculature supplies oxygen to the inner layers of the retina, the light-sensitive tissue in the eye. During development, formation of the retinal vasculature depends on prior establishment of a mesh of astrocytes, a type of glial cell, which guide the growth of the vascular network. Astrocytes emerge from the optic nerve head and proliferate and spread, forming a mesh-like layer over the retinal surface. The initially formed cells are termed astrocyte precursor cells (APCs), which differentiate into immature perinatal astrocytes (IPAs) during the prenatal period. A continuum model is developed to describe the proliferation, differentiation, and migration these cells. Effects of oxygen and growth factor levels on proliferation and differentiation are included. Cell migration is driven by gradients in tension in the astrocyte mesh, which varies inversely with total density. The resulting governing equations have the form of a nonlinear diffusion-like equation. The model can account for the observed radial spread over time of the astrocyte disk. Experimental observations show that the APCs form a narrow rim around the edge of this disk, with IPAs in the interior. The model predicts this behavior if the mobility of the APCs is assumed to be higher than that of the IPAs under a given tension gradient. Thus, the model shows how tension-driven cell motions can account for separation of cell types in a cell layer spreading over a substrate.

Long-term cooperation, competition, or exploitation among individuals can be modeled through repeated games. In repeated games, Press and Dyson discovered zero-determinant (ZD) strategies that enforce a special relationship between two players. This special relationship implies that a ZD player can unilaterally impose a linear payoff relationship to the opponent regardless of the opponent's strategies. A ZD player also has a property that can lead the opponent to an unconditional cooperation if the opponent tries to improve its payoff. This property has been mathematically confirmed by Chen and Zinger. Humans often underestimate a payoff obtained in the future. However, such discounting was not considered in their analysis. Here, we mathematically explored whether a ZD player can lead the opponent to an unconditional cooperation even if a discount factor is incorporated. Specifically, we represented the expected payoff with a discount factor as the form of determinants and calculated whether the values obtained by partially differentiating each factor in the strategy vector become positive. As a result, we proved that the strategy vector ends up as an unconditional cooperation even when starting from any initial strategy. This result was confirmed through numerical calculations. We extended the applicability of ZD strategies to real world problems.

Several local outbreaks have occurred and been suppressed with the dynamic zero-COVID policy and widely promoted vaccination program implemented in China. The epidemic duration and final size vary significantly in different cities, which may be attributed to different implementation patterns and intensities of non-pharmaceutical interventions (NPIs). It's important to capture the underlying mechanism to explore more efficient implementation patterns of NPIs in order to prevent the future resurgence. In this study, outbreaks caused by Delta variant in Xi'an, Yangzhou and Guangzhou in 2021 are chosen as the examples. A novel model dividing the population into three groups is proposed to describe the heterogeneity of control interventions. The model is calibrated and key parameters related to NPIs are estimated by using multi-source epidemic data. The estimation results show a lower transmission probability but a higher initial reproduction number in Xi'an. Sensitivity analysis are conducted to investigate the impact of various control measures in different epidemic phases. The results identify the vital role of enhancing closed-off management, strengthening tracing and testing intensities, on shortening the epidemic durations and reducing the final size. Further, we find that sufficiently implemented closed-off management would prevent the city from lockdown. Strengthening the tracing other than the testing strategy in the initial stage is more effective on containing the epidemic in a shorter duration with less infections.

Invasive species account for incalculable damages worldwide, in both ecological and bioeconomic terms. The question of how a network of invasive populations can be optimally managed is one that deserves further exploration. A study accounting for partial observability and imperfect detection, in particular, could yield useful insights into species eradication efforts. Here, we generalized a simple model system that we developed in previous work. This model consists of three interacting populations with underlying strong Allee effects and stochastic dynamics, inhabiting distinct locations connected by dispersal, which can generate bistability. To explore the stochastic dynamics, we formulated an individual-based modeling approach. Next, using the theory of continuous-time Markov chains, we approximated the original high-dimensional model by a Markov chain with eight states, with each state corresponding to a combination of population thresholds. We then used the reduced model as the core for a powerful decision-making tool, referred to as a Partially Observable Markov Decision Process (POMDP). Analysis of this POMDP indicates when the system results in optimal management outcomes.

Phase variation, which causes splitting the homogeneous bacterial population into subpopulations of the cells with different and reversible activity of the same variable genes, provide bacteria with wide opportunities to implement their adaptation strategies. Meanwhile, the gene networks perform integral regulation of the genome and control over the formation of phenotypic traits of the organism. Mechanisms of phase variation, such as genomic rearrangements and DNA methylation, affect the regulatory contours of gene networks. At the same time, the gene networks themselves may be the key cause of the appearance of phenotypic variants. Here we propose a new class of gene networks that represent a ring connection of genetic triggers based on an oscillator with cyclic gene repression, termed the tringers. It is shown in silico that the tringers are able to provide splitting the homogeneous bacterial population into cellular subpopulations with alternative expression of controlled genes (phases), stable inheritance of phases, and switching between them, in particular, returning to a homogeneous state. Switching can be carried out under the influence of external and internal factors, both directly and by signals from specialized molecular genetic systems of the cell, in particular, the response to various types of stress. Such epigenetic structures with the function of a phase switching controller are possible in nature and applicable in synthetic biology.

One of the fundamental mechanisms in embryogenesis is the process by which cells differentiate and create tissues and structures important for functioning as a multicellular organism. Morphogenesis involves diffusive process of chemical signalling involving morphogens that pre-pattern the tissue. These morphogens influence cell fate through a highly nonlinear process of transcriptional signalling. In this paper, we consider this multiscale process in an idealised model for a growing domain. We focus on intracellular processes that lead to robust differentiation into two cell lineages through interaction of a single morphogen species with a cell fate variable that undergoes a bifurcation from monostability to bistability. In particular, we investigate conditions that result in successful and robust pattern formation into two well-separated domains, as well as conditions where this fails and produces a pinned boundary wave where only one part of the domain grows. We show that successful and unsuccessful patterning scenarios can be characterised in terms of presence or absence of a folded saddle singularity for a system with two slow variables and one fast variable; this models the interaction of slow morphogen diffusion, slow parameter drift through bifurcation and fast transcription dynamics. We illustrate how this approach can successfully model acquisition of three cell fates to produce three-domain "French flag" patterning, as well as for a more realistic model of the cell fate dynamics in terms of two mutually inhibiting transcription factors.

Stochastic individual-based mathematical models are attractive for modelling biological phenomena because they naturally capture the stochasticity and variability that is often evident in biological data. Such models also allow us to track the motion of individuals within the population of interest. Unfortunately, capturing this microscopic detail means that simulation and parameter inference can become computationally expensive. One approach for overcoming this computational limitation is to coarse-grain the stochastic model to provide an approximate continuum model that can be solved using far less computational effort. However, coarse-grained continuum models can be biased or inaccurate, particularly for certain parameter regimes. In this work, we combine stochastic and continuum mathematical models in the context of lattice-based models of two-dimensional cell biology experiments by demonstrating how to simulate two commonly used experiments: cell proliferation assays and barrier assays. Our approach involves building a simple statistical model of the discrepancy between the expensive stochastic model and the associated computationally inexpensive coarse-grained continuum model. We form this statistical model based on a limited number of expensive stochastic model evaluations at design points sampled from a user-chosen distribution, corresponding to a computer experiment design problem. With straightforward design point selection schemes, we show that using the statistical model of the discrepancy in tandem with the computationally inexpensive continuum model allows us to carry out prediction and inference while correcting for biases and inaccuracies due to the continuum approximation. We demonstrate this approach by simulating a proliferation assay, where the continuum limit model is the well-known logistic ordinary differential equation, as well as a barrier assay where the continuum limit model is closely related to the well-known Fisher-KPP partial differential equation. We construct an approximate likelihood function for parameter inference, both with and without discrepancy correction terms. Using maximum likelihood estimation, we provide point estimates of the unknown parameters, and use the profile likelihood to characterise the uncertainty in these estimates and form approximate confidence intervals. For the range of inference problems considered, working with the continuum limit model alone leads to biased parameter estimation and confidence intervals with poor coverage. In contrast, incorporating correction terms arising from the statistical model of the model discrepancy allows us to recover the parameters accurately with minimal computational overhead. The main tradeoff is that the associated confidence intervals are typically broader, reflecting the additional uncertainty introduced by the approximation process. All algorithms required to replicate the results in this work are written in the open source Julia language and are available at GitHub.

We introduce the term net-proliferation rate for a class of harvested single species models, where harvest is assumed to reduce the survival probability of individuals. Following the classical maximum sustainable yield calculations, we establish relations between the proliferation and net-proliferation that are economically and sustainably favored. The resulting square root identities are analytically derived for species following the Beverton-Holt recurrence considering three levels of complexity. To discuss the generalization of the results, we compare the square root result to the optimal survival rate of the Pella-Tomlinson model. Furthermore, to test the practical relevance of the square root identities, we fit a stochastic Pella-Tomlinson model to observed Barramundi fishery data from the Southern Gulf of Carpentaria, Australia. The results show that for the estimated model parameters, the equilibrium biomass levels resulting from the MSY harvest and the square root harvest are similar, supporting the claim that the square root harvest can serve as a rule-of-thumb. This application, with its inherited model uncertainty, sparks a risk sensitivity analysis regarding the probability of populations falling below an unsustainable threshold. Characterization of such sensitivity helps in the understanding of both dangers of overfishing and potential remedies.

Non Small Cell Lung Cancer (NSCLC) is the most common type of lung cancer, and represents the leading cause of cancer-related deaths worldwide. Experimental studies have shown that these solid cancers are heavily infiltrated with macrophages: anti-tumour M1 macrophages, pro-tumour M2 macrophages, and macrophage subtypes sharing both M1 and M2 properties. In this study we aim to investigate qualitatively the role of macrophages with different functional phenotypes (especially those with mixed phenotypes) on cancer dynamics and the success of different immunotherapies for cancer. To this end, we start with two time-evolving mathematical models for cancer-immune interactions that consider: (i) the effect of the two extreme phenotypes, M1 and M2 cells; (ii) the effect of M1 and M2 cells, as well as a macrophage sub-population with a mixed phenotype (throughout this theoretical study we call these cells "M12 cells"). We compare the dynamics of the two models using computational approaches, paying particular attention to the effect of different anti-cancer immunotherapies that focus on macrophages. Since data available for NSCLC and macrophage interactions are incomplete, we perform a global sensitivity analysis to see the influence of input parameters on model outcomes. Finally, we consider extensions of the previous two models to include also the spatial movement of cells, and investigate the role of macrophages with extreme phenotypes and with mixed phenotypes, on the invasion of cancer cells into the surrounding extracellular matrix (ECM). We use numerical simulations to investigate the macrophages phenotypes at the tumour center versus the invasive margin. Again, we examine the impact of immunotherapies for cancer on the spatial dynamics of cancers and immune cells, and observe a shift in the phenotype of macrophages distributed at the tumour center and invasive margin.