Equilibrium Population Distribution Research Articles

Abstract B010: Ecological epistasis can drastically alter evolutionary trajectories on genotypic fitness landscapes, maintaining, masking or mimicking genetic selection in evolutionary simulations

Abstract The evolution of drug resistance is broadly understood in terms of the evolution and survival of the genotype that is "fittest" under treatment. There are many computational evolutionary models built upon the assumption that under treatment, cancer cells with the fittest genotype persist and ultimately prevail. Many of these models lack sufficient complexity to predict tumor evolution in the clinic. In particular, the standard Wright-Fisher approach taken from population genetics neither considers nor accurately predicts the heterogeneity that we see in patients and cell-cell interactions are not included. To combat this we develop a model of evolution that incorporates average cell-cell interactions by adding game theoretic interactions to a Wright-Fisher model with mutation and use this to explore and model cases of interest. In our agent-based model we start with N wildtype cells and represent the genotype of each cell with a binary string of n alleles. We represent mutations to the genotype as transitions from 0 to 1 and vice-versa in an allele and each unique genotype is a cell type. Each cell in the non-spatial, well-mixed agent-based model can be selected to reproduce with probability p proportional to the fitness of its genotype and each daughter cell can undergo mutation with probability μ. At each new time step we modify the fitness of each cell type, using the multiplication of the payoff matrix with the cell type frequency vector to reflect the average fitness benefits and costs of cell-cell interactions. We vary the payoff matrices for up to 4 alleles (16 possible genotypes) and show how the addition of these ecological game interactions between different cell populations can modify evolutionary trajectories. We find, for example, that with certain strengths of game interactions, the evolution on an initially selective genotypic landscape can mimic neutral landscapes and vice versa. We term this addition of ecological game interactions to genotypic landscapes "ecological epistasis". We derive mathematical expressions for the equilibrium population distribution in the single allele (two genotype) with arbitrary games case. We also derive expressions for the payoff matrices required to mimic selective games and to make selective landscapes neutral. Furthermore we derive an expression for the curvature of the resultant population distribution, producing a "game signature” with which to probe temporal evolutionary data for the presence of games. This model emphasizes that the emergence of a dominant cell-type can be caused by both ecological and genotypic factors and conversely that the genotypic landscape does not need to be neutral for heterogeneity to persist. Due to the multiclonal and heterogeneous nature of tumors and their microenvironment, these results are of key importance in considering treatment approaches and interpreting existing patient data. We highlight how game interactions may explain heterogeneous patient outcomes and contradicting results on the fitness of cancer genotypes in the literature. Citation Format: Steph J. Owen, Michael Hinczewski, Jacob G. Scott. Ecological epistasis can drastically alter evolutionary trajectories on genotypic fitness landscapes, maintaining, masking or mimicking genetic selection in evolutionary simulations [abstract]. In: Proceedings of the AACR Special Conference on the Evolutionary Dynamics in Carcinogenesis and Response to Therapy; 2022 Mar 14-17. Philadelphia (PA): AACR; Cancer Res 2022;82(10 Suppl):Abstract nr B010.

Diffusive Resettlement: Irreversible Urban Transitions in Closed Systems.

We propose a non-equilibrium framework for modelling the evolution of cities, which describes intra-urban migration as an irreversible diffusive process. We validate this framework using the actual migration data for the Australian capital cities. With respect to the residential relocation, the population is shown to be composed of two distinct groups, exhibiting different relocation frequencies. In the context of the developed framework, these groups can be interpreted as two components of a binary fluid mixture, each with its own diffusive relaxation time. Using this approach, we obtain long-term predictions of the cities’ spatial structures, which define their equilibrium population distribution.

Precautionary public health, ageing and urban agglomeration

Introducing precautionary public health policy into an overlapping generations model with migration of households between regions, we discuss the causality between ageing and urban agglomeration. We analyze the effect of public policy to expand longevity on an equilibrium population distribution between an urban region and a rural region in a steady state. As the result of our analysis, it is possible that the promotion of precautionary public health policy leads to enlarge wage differentials and enlarge regional disparity.

Kinetic network study of the diversity and temperature dependence of Trp-Cage folding pathways: combining transition path theory with stochastic simulations.

We present a new approach to study a multitude of folding pathways and different folding mechanisms for the 20-residue mini-protein Trp-Cage using the combined power of replica exchange molecular dynamics (REMD) simulations for conformational sampling, transition path theory (TPT) for constructing folding pathways, and stochastic simulations for sampling the pathways in a high dimensional structure space. REMD simulations of Trp-Cage with 16 replicas at temperatures between 270 and 566 K are carried out with an all-atom force field (OPLSAA) and an implicit solvent model (AGBNP). The conformations sampled from all temperatures are collected. They form a discretized state space that can be used to model the folding process. The equilibrium population for each state at a target temperature can be calculated using the weighted-histogram-analysis method (WHAM). By connecting states with similar structures and creating edges satisfying detailed balance conditions, we construct a kinetic network that preserves the equilibrium population distribution of the state space. After defining the folded and unfolded macrostates, committor probabilities (P(fold)) are calculated by solving a set of linear equations for each node in the network and pathways are extracted together with their fluxes using the TPT algorithm. By clustering the pathways into folding "tubes", a more physically meaningful picture of the diversity of folding routes emerges. Stochastic simulations are carried out on the network, and a procedure is developed to project sampled trajectories onto the folding tubes. The fluxes through the folding tubes calculated from the stochastic trajectories are in good agreement with the corresponding values obtained from the TPT analysis. The temperature dependence of the ensemble of Trp-Cage folding pathways is investigated. Above the folding temperature, a large number of diverse folding pathways with comparable fluxes flood the energy landscape. At low temperature, however, the folding transition is dominated by only a few localized pathways.

Thermodynamic Analysis of Interacting Nucleic Acid Strands

Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands. This dynamic program is based on a rigorous extension of secondary structure models to the multistranded case, addressing representation and distinguishability issues that do not arise for single‐stranded structures. We then derive the form of the partition function for a fixed volume containing a dilute solution of nucleic acid complexes. This expression can be evaluated explicitly for small numbers of strands, allowing the calculation of the equilibrium population distribution for each species of complex. Alternatively, for large systems (e.g., a test tube), we show that the unique complex concentrations corresponding to thermodynamic equilibrium can be obtained by solving a convex programming problem. Partition function and concentration information can then be used to calculate equilibrium base‐pairing observables. The underlying physics and mathematical formulation of these problems lead to an interesting blend of approaches, including ideas from graph theory, group theory, dynamic programming, combinatorics, convex optimization, and Lagrange duality.

Error-threshold exists in fitness landscapes with lethal mutants

BackgroundOne of the important insights of quasi-species theory is an error-threshold. The error-threshold is the error rate of replication above which the sudden onset of the population delocalization from the fittest genotype occurs despite Darwinian selection; i.e., the break down of evolutionary optimization. However, a recent article by Wilke in this journal, after reviewing the previous studies on the error-threshold, concluded that the error-threshold does not exist if lethal mutants are taken into account in a fitness landscape. Since lethal mutants obviously exist in reality, this has a significant implication about biological evolution. However, the study of Wagner and Krall on which Wilke's conclusion was based considered mutation-selection dynamics in one-dimensional genotype space with the assumption that a genotype can mutate only to an adjoining genotype in the genotype space. In this article, we study whether the above conclusion holds in high-dimensional genotype space without the assumption of the adjacency of mutations, where the consequences of mutation-selection dynamics can be qualitatively different.ResultsTo examine the effect of mutant lethality on the existence of the error-threshold, we extend the quasi-species equation by taking the lethality of mutants into account, assuming that lethal genotypes are uniformly distributed in the genotype space. First, with the simplification of neglecting back mutations, we calculate the error-threshold as the maximum allowable mutation rate for which the fittest genotype can survive. Second, with the full consideration of back mutations, we study the equilibrium population distribution and the ancestor distribution in the genotype space as a function of error rate with and without lethality in a multiplicative fitness landscape. The results show that a high lethality of mutants actually introduces an error-threshold in a multiplicative fitness landscape in sharp contrast to the conclusion of Wilke. Furthermore, irrespective of the lethality of mutants, the delocalization of the population from the fittest genotype occurs for an error rate much smaller than random replication. Finally, the results are shown to extend to a system of finite populations.ConclusionHigh lethality of mutants introduces an error-threshold in a multiplicative fitness landscape. Furthermore, irrespective of the lethality of mutants, the break down of evolutionary optimization happens for an error rate much smaller than random replication.

Tiebout equilibria in local public good economies with spillovers

This paper analyzes the effects of spillovers on the equilibrium population distribution across jurisdictions in a local public good economy with free mobility. Spillovers are parametrized by a matrix [ α ij ] where α ij ∈ [0, 1]. When spillovers are symmetric and close to 0 or 1 (pure local public goods and pure public goods), all equilibrium jurisdiction structures are symmetric. However, any population distribution can be sustained in equilibrium for some value of the spillover parameter α. In the class of utility functions with additive externalities, we identify the unique family of utility functions for which equilibria are symmetric except for an isolated value of α. This is a class of utility functions which are linear in the public good and a power function of the private good, u( c, γ) = − A(1 − c) β + γ. With this specification of utility, we show that an increase in α results in a more fragmented equilibrium population distribution, and that when spillovers are asymmetric and large, a jurisdiction which is more centrally located (i.e. benefits more from the public goods provided in other jurisdictions) has a larger population in equilibrium.

Investigation of the Chromatographic Process via Pulsed-Gradient Spin−Echo Nuclear Magnetic Resonance. Role of the Solvent Composition in Partitioning Chromatography

The diffusional properties of molecules in solution vary dramatically upon addition of a solid chromatographic phase. This effect can be monitored via pulsed-gradient spin-echo NMR used in conjunction with moderately fast rotation of the sample (high-resolution magic angle spinning) to produce exploitable spectra. The molecular diffusion coefficients observed in this condition are averages reflecting the equilibrium population distribution among the different phases. It is thus possible to use this information for investigating a crucial step of reversed-phase chromatography, namely, the partitioning of the analyte between different phases. In this work, we describe the evolution of the apparent diffusion coefficient of typical solutes for water/acetonitrile solvent mixtures of varying proportions.

Movement effects on equilibrium distributions of habitat generalists in heterogeneous landscapes

The effect of landscape heterogeneity on population distribution and persistence has been well investigated for habitat specialists, but the response of habitat generalist to landscape heterogeneity is less well known. We used a matrix model for an agricultural habitat generalist carabid ( Pterostichus cupreus (L.)) on a lattice landscape, to study the effect of changing landscape composition and configuration on the within generation equilibrium (asymptotic) population distribution. Movements were approximated from diffusion functions that depended on habitat quality only. The population distribution of P. cupreus was sensitive to both habitat composition and configuration. Habitat configuration generally explained more variation in the population distribution relative the resources, but the changes in amount of preferred habitat had a larger effect at low amounts of preferred habitat. The resource use of P. cupreus was less sensitive when there was low contrast between habitat qualities. Numerical solutions indicated that the stable population distribution is usually reached within a generation, and the analytical results from our equilibrium model are thus reasonable approximations. The (transient) time to reach equilibrium population distribution was lower in landscapes where preferred habitat was scarce and scattered, and there was a trade-off between transient time and the population distribution relative to the resources. We found no clear threshold effects, only a gradually steeper decline in resource use as preferred resources were randomly lost in high contrast landscapes. Overall, the results were congruent with other results on generalists where demography and density dependent processes have been included, which indicate that movement alone is a driving force.

Tiebout Equilibria in Local Public Good Economies with Spillovers

This paper analyzes the effects of spillovers on the formation of jurisdictions in a local public good economy with free mobility. The number of jurisdictions is fixed and spillovers are parametrized by a matrix [αij where αij e [0,1]. When spillovers are symmetric and close to 0 or 1 (pure local public goods and pure public goods), all equilibrium jurisdiction structures are symmetric. However, any population distribution can be sustained in equilibrium for some value of the spillover parameter α. In the class of quasi-linear utility functions, we identify the unique family of utility functions for which equilibria are symmetric except for an isolated value of α. This is a class of utility functions which are linear in the public good and a power function of the private good, u(c, γ) = -A(1-c)β+ γ. With this specification of utility, we show that an increase in α results in a more fragmented equilibrium population distribution, and that when spillovers are asymmetric and large, a jurisdiction which is more centrally located (i.e. benefits more from the public goods provided in other jurisdictions) has a larger population in equilibrium.

Pressure effects on bimolecular recombination and unimolecular dissociation reactions

The treatment of pressure effects on bimolecular recombinations and unimolecular dissociations is discussed. The analysis of recombination and dissociation reactions is made by showing how the nonequilibrium energy (E) and angular momentum (J)-dependent steady-state population distribution functions for the two reactions are related to each other and to the equilibrium population distribution function at the given E and J. As a special case a strong collision model is then used for the collisional rotational angular momentum transfer, and a ladder model for the collisional energy transfer. An analytical result is obtained for states below the dissociation threshold. The extension to recombinations with two exit channels is described, for application to ozone formation and isotopic effects.

The distribution and abundance of populations limited at multiple spatial scales

Summary 1. We use mathematical programming to explore long‐term equilibrium population distribution and abundance effects arising from a single‐patch reaction–diffusion model when ecological factors limit carrying capacity at two different spatial scales. Both homoogeneous and heterogeneous habitat patches are investigated under passive diffusion conditions. Capacity limits, reproduction and dispersal rates, and habitat preferences are individually varied. 2. When landscape‐scale habitat factors are strongly limiting, and reproduction and dispersal processes are less so, suitable population arrangements can be either clustered or highly fragmented. 3. Population clustering also occurs when strong habitat preferences are introduced, even though few members of the clusters may actually occupy the preferred sites. 4. As reproduction or dispersal rates become increasingly limiting in the system (i.e. as the parameters approach extinction thresholds), population clustering within a habitat patch is required to reach long‐term equilibrium population levels. 5. These results offer a potential explanation for why species–habitat association studies have been characterized by high variability.

Realistic master equation modeling of relaxation on complete potential energy surfaces: Partition function models and equilibrium results

To elucidate the role that potential surface topography plays in shaping the evolution of a cluster toward equilibrium, entire sets of kinetically accessible bound-state configurations and transition states on the model potential energy surfaces of (KCl)5 and Ar9 are mapped and compared. To describe the stochastic dynamics on these surfaces in terms of transition-state theory, we require adequate approximations of the partition functions of the minima and transition states. In this paper we introduce several partition function models derived from harmonic and anharmonic approximations and compare their predicted equilibrium population distributions with those determined from canonical-ensemble molecular dynamics. We perform this comparison for both (KCl)5 and Ar9 in order to evaluate the relative performance of the models for two different types of potential surfaces. For each system, particular models are found to give results that agree better with simulation than do the results using the simple harmonic approximation. However, no one unparameterized model gives acceptable results for all minima, and the best parameter-free strategies differ for (KCl)5 and Ar9. Nevertheless, a one-parameter version of one of the models is shown to give the best agreement with simulation for both systems. In an accompanying paper, the best partition function models are used to construct a stochastic master equation which makes predictions of relaxation behavior. These predictions are compared with results from molecular dynamics.

Effect of the Bloch - Siegert shift in a strongly driven transition: asymmetric Autler - Townes profile

Under the conditions of the rotating wave approximation (RWA), a transition strongly driven by a resonant oscillating field displays the well known symmetric Autler - Townes doublet. However, if the counter-rotating component, neglected in the RWA, is taken into account, the Bloch - Siegert shift gives rise to an Autler - Townes doublet of unequal intensity even in the case of a resonant driving field. This effect is investigated theoretically in a V-shaped three-level double-resonance configuration and the results are presented in this paper. An interesting observation is that the level of asymmetry not only depends on the driving-field intensity but also on the characteristics of the driven system including relaxation rates and equilibrium population distributions.

A dynamic population model incorporating a variety of urban functions.

"This paper aims to investigate the theoretical background of Allen and Sanglier's model, slightly modifying it. We construct a dynamic two-city population model in which there are different types of urban functions in the cities. Population dynamics of both cities are expressed in terms of a system of differential equations of the logistic type, taking account of spatial interaction between them. For this model, both the existence condition and the globally asymptotic stability of the equilibrium state are theoretically explored. Then configurations of equilibrium population distribution that correspond to different combinations of the numbers of types of urban functions in two cities are also examined. Finally, some numerical simulations are done in order to supplement the theoretical considerations."

An Analysis of Migration Streams for the Canadian Regional System, 1952–1983: 2. Disequilibrium

This is part two of a two‐part series that seeks to specify the factors and processes contributing to aggregate change in the Canadian population distribution over the recent past. In part one, we use a model of migration probabilities to identify socioeconomic factors that explain the observed interregional migration flows. In part two, we use an adjustment process for our regional population system to calculate equilibrium population distributions. The estimated migration probabilities of part one are essential for this calculation, thus linking equilibria with the socioeconomic factors previously identified. Disequilibrium, defined as the difference between observed and equilibrium population distributions, is then used to infer relative growth and decline tendencies in the system. We examine and justify our results in the larger context of contemporary Canadian history.

An Analysis of Migration Streams for the Canadian Regional System, 1952–1983 1. Migration Probabilities

This is part one of a two‐part series that seeks to specify the factors and processes contributing to aggregate change in the Canadian population distribution over the recent past. In part one, we use a model of migration probabilities to identify socioeconomic factors that explain the observed interregional migration flows. In part two, we use an adjustment process for our regional population system to calculate equilibrium population distributions. The estimated migration probabilities of part one are essential for this calculation, thus linking equilibria with the socioeconomic factors previously identified. Disequilibrium, defined as the difference between observed and equilibrium population distributions, is then used to infer relative growth and decline tendencies in the system. We examine and justify our results in the larger context of contemporary Canadian history.

Collisional energy transfer at high temperatures from the biased walk model

A new expression for the probability, P(E,E'), of collisional energy transfer for large polyatomics at high temperatures is deduced by using the assumptions of the biased random walk (BRW) model [Lim, K. F.; Gilbert, R. G. J. Chem. Phys. 1986, 84, 6124] and of a Gaussian equilibrium population distribution. The latter approximation is shown to be very accurate at sufficiently high temperatures. Simple analytic expressions are obtained for P(E,E'), which show the correct limiting weak- and strong-collision forms as the duration of a collision approaches zero and infinity, respectively. Expressions are found for the average energy-transfer quantities , , and , as functions of initial energy E'. The expression for explicitly shows the change of sign as E' is varied over a wide range of E'. The new BRW expression for P(E,E') can be accurately approximated as Gaussian. The derived expressions will be of use in modeling high-temperature processes such as combustion.

On the foundations of land use theory: Discrete versus continuous populations

Urban economists and location theorists have long employed land use models with a continuum of agents distributed over a continuum of locations. However, these continous models have been criticized on behavioral grounds in that individual households can consume only zero amounts of land in equilibrium. Hence the central purpose of this paper is to propose an alternative interpretation of these continous models as limiting approximations of discrete population models. In particular, it is shown that for large population sizes, the population distributions of the classical continuous model uniformly approximate the equilibrium population distributions generated by an appropriately defined class of discrete population models.

Conformational analysis and molecular dynamics simulations of maltose.

AbstractConstrained conformational energy minimizations have been used to calculate an adiabatic (Φ, ψ) potential energy surface for the disaccharide β‐maltose. The inclusion of molecular flexibility in the conformational energy analysis of the disaccharide was found to significantly lower the barriers to conformational transitions, as has been observed previously for other systems. Several low energy wells were identified on the adiabatic surface which differ in energy by small amounts and with low absolute barriers separating them, indicating the possibility of a non‐negligible equilibrium population distribution in each well. If such a distribution of conformations existed in the physical system, the conformation observed by NMR NOE measurements would thus be a “virtual” conformation. Molecular dynamics simulations of the motions of this molecule in vacuum were also conducted and indicate that the rate of relaxation of the molecule to the adiabatic surface may be slower than the typical timescale of conformational fluctuations. This effect is apparently due to an unphysical persistence of hydrogen bond patterns in vacuum which does not occur in aqueous solution. Trajectories undergoing transitions between wells were calculated and the effects of such conformational transitions upon the ensemble mean structure, such as might be observed in an NMR experiment, were demonstrated.