A stochastic gene evolution model with time dependent mutations

We develop here a new class of gene evolution models in which the nucleotide mutations are time dependent. These models allow to study nonlinear gene evolution by accelerating or decelerating the mutation rates at different evolutionary times. They generalize the previous ones which are based on constant mutation rates. The stochastic model developed in this class determines at some time t the occurrence probabilities of trinucleotides mutating according to 3 time dependent substitution parameters associated with the 3 trinucleotide sites. Therefore, it allows to simulate the evolution of the circular code recently observed in genes. By varying the class of function for the substitution parameters, 1 among 12 models retrieves after mutation the statistical properties of the observed circular code in the 3 frames of actual genes. In this model, the mutation rate in the 3rd trinucleotide site increases during gene evolution while the mutation rates in the 1st and 2nd sites decrease. This property agrees with the actual degeneracy of the genetic code. This approach can easily be generalized to study evolution of motifs of various lengths, e.g., dicodons, etc., with time dependent mutations.     

A stochastic model of gene evolution with chaotic mutations

We develop here a new class of stochastic models of gene evolution in which the mutations are chaotic, i.e. a random subset of the 64 possible trinucleotides mutates at each evolutionary time t according to some substitution probabilities. Therefore, at each time t, the numbers and the types of mutable trinucleotides are unknown. Thus, the mutation matrix changes at each time t. The chaotic model developed generalizes the standard model in which all the trinucleotides mutate at each time t. It determines the occurrence probabilities at time t of trinucleotides which chaotically mutate according to three substitution parameters associated with the three trinucleotide sites. Two theorems prove that this chaotic model has a probability vector at each time t and that it converges to a uniform probability vector identical to that of the standard model. Furthermore, four applications of this chaotic model (with a uniform random strategy for the 64 trinucleotides and with a particular strategy for the three stop codons) allow an evolutionary study of the three circular codes identified in both eukaryotic and prokaryotic genes. A circular code is a particular set of trinucleotides whose main property is the retrieval of the frames in genes locally, i.e. anywhere in genes and particularly without start codons, and automatically with a window of a few nucleotides. After a certain evolutionary time and with particular values for the three substitution parameters, the chaotic models retrieve the main statistical properties of the three circular codes observed in genes. These applications also allow an evolutionary comparison between the standard and chaotic models.     

A stochastic model for adhesion-mediated cell random motility and haptotaxis

The active migration of blood and tissue cells is important in a number of physiological processes including inflammation, wound healing, embryogenesis, and tumor cell metastasis. These cells move by transmitting cytoplasmic force through membrane receptors which are bound specifically to adhesion ligands in the surrounding substratum. Recently, much research has focused on the influence of the composition of extracellular matrix and the distribution of its components on the speed and direction of cell migration. It is commonly believed that the magnitude of the adhesion influences cell speed and/or random turning behavior, whereas a gradient of adhesion may bias the net direction of the cell movement, a phenomenon known as haptotaxis. The mechanisms underlying these responses are presently not understood.A stochastic model is presented to provide a mechanistic understanding of how the magnitude and distribution of adhesion ligands in the substratum influence cell movement. The receptor-mediated cell migration is modeled as an interrelation of random processes on distinct time scales. Adhesion receptors undergo rapid binding and transport, resulting in a stochastic spatial distribution of bound receptors fluctuating about some mean distribution. This results in a fluctuating spatio-temporal pattern of forces on the cell, which in turn affects the speed and turning behavior on a longer time scale. The model equations are a system of nonlinear stochastic differential equations (SDE's) which govern the time evolution of the spatial distribution of bound and free receptors, and the orientation and position of the cell. These SDE's are integrated numerically to simulate the behavior of the model cell on both a uniform substratum, and on a gradient of adhesion ligand concentration.Furthermore, analysis of the governing SDE system and corresponding Fokker-Planck equation (FPE) yields analytical expressions for indices which characterize cell movement on multiple time scales in terms of cell cytomechanical, morphological, and receptor binding and transport parameters. For a uniform adhesion ligand concentration, this analysis provides expressions for traditional cell movement indices such as mean speed, directional persistence time, and random motility coefficient. In a small gradient of adhesion, a perturbation analysis of the FPE yields a constitutive cell flux expression which includes a drift term for haptotactic directional cell migration. The haptotactic drift contains terms identified as contributions from directional orientation bias (taxis).     

On the increase of chromosome mutations under random mating.

After a short introduction on karyotypes and chromosome mutations, we review the ways by which a chromosome mutation can increase in a random mating population, despite the mutation's deleterious effect on the fertility of heterozygotes. Random drift, segregation distortion, viability advantage, and recombination modification are the mechanisms considered. When possible, the models are illustrated with examples of chromosome mutations involving autosomes in mammals, but the arguments apply, of course, to any genetic factor in any outbreeding species that causes a fertility decrease in heterozygotes.     

A calcium model with random absorption: a stochastic approach.

Absorption of calcium, or any mineral, by the body is subject to the random fluctuations typical of diffusion through membranes. In this paper we consider the absorption of calcium from the gut as a white noise process added to the deterministic model of Sen & Mohr (1990, J. theor. Biol. 142, 179-188). The first two moments for the amount of calcium in the extracellular fluid (ECF) have been derived using the Ito Calculus. A confidence interval for the total amount of calcium in the ECF is constructed. The equations for the first two moments of the fraction of dose calcium in the ECF are also given. Suggestions are made for the collection of experimental data in a form which should be helpful in investigating the magnitude of the stochastic effect.     

Role of gestation delay in a plankton-fish model under stochastic fluctuations

The present paper studies a minimal prey-predator model in the context of marine plankton interaction together with predation by planktivorous fish. The time lag required for gestation of the predator is incorporated and the resulting delayed model is analyzed for stability and bifurcation phenomena. A stochastic extension of the model is considered by perturbing the growth process of phytoplankton using colored noise process known to be more appropriate for the marine environment. The stochastic models with and without gestation delay are analyzed for stability aspects and a threshold value of gestation delay is obtained; this threshold is then compared with that of the deterministic model.     

Quantification of random genomic mutations

Cancer cells contain numerous clonal mutations. It has been theorized that malignant cells sustain an elevated mutation rate and, as a consequence, harbor yet larger numbers of random point mutations. Testing this hypothesis has been precluded by lack of an assay to measure random mutations-that is, mutations that occur in only one or a few cells of a population. We have established a method that has permitted us to detect and identify rare random mutations in human cells, at a frequency of 1 per 10(8) base pairs. The assay is based on gene capture, by hybridization with a uracil-containing probe, followed by magnetic separation. Mutations that render the mutational target sequence non-cleavable by a restriction enzyme are quantified by dilution to single molecules and real-time quantitative PCR amplification. The assay can be extended to quantify mutation in any DNA-based organism, at different sites in the genome, in introns and exons, in unselected and selected genes, and in proliferating and quiescent cells.     

A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations

Two central features of polymorphonuclear leukocyte chemosensory movement behavior demand fundamental theoretical understanding. In uniform concentrations of chemoattractant, these cells exhibit a persistent random walk, with a characteristic "persistence time" between significant changes in direction. In chemoattractant concentration gradients, they demonstrate a biased random walk, with an "orientation bias" characterizing the fraction of cells moving up the gradient. A coherent picture of cell movement responses to chemoattractant requires that both the persistence time and the orientation bias be explained within a unifying framework. In this paper, we offer the possibility that "noise" in the cellular signal perception/response mechanism can simultaneously account for these two key phenomena. In particular, we develop a stochastic mathematical model for cell locomotion based on kinetic fluctuations in chemoattractant/receptor binding. This model can simulate cell paths similar to those observed experimentally, under conditions of uniform chemoattractant concentrations as well as chemoattractant concentration gradients. Furthermore, this model can quantitatively predict both cell persistence time and dependence of orientation bias on gradient size. Thus, the concept of signal "noise" can quantitatively unify the major characteristics of leukocyte random motility and chemotaxis. The same level of noise large enough to account for the observed frequency of turning in uniform environments is simultaneously small enough to allow for the observed degree of directional bias in gradients.     

The probability of causation under a stochastic model for individual risk

In this paper we offer a mathematical definition for the probability of causation that formalizes the legal and ordinary-language meaning of the term. We show that, under this definition, even the average probability of causation among exposed cases is not identifiable from epidemiologic data. This is because the probability of causation depends both on the unknown mechanisms by which exposure affects disease risk and competing risks, and on the unknown degree of heterogeneity in the background disease risk of the exposed population. We derive the maximum and minimum values for the probability of causation consistent with the observable population quantities. We also derive the relationship of the "assigned share" (excess incidence rate as a proportion of total incidence rate) to the probability of causation.     

Selection on plasticity of seasonal life-history traits using random regression mixed model analysis

Theory considers the covariation of seasonal life-history traits as an optimal reaction norm, implying that deviating from this reaction norm reduces fitness. However, the estimation of reaction-norm properties (i.e., elevation, linear slope, and higher order slope terms) and the selection on these is statistically challenging. We here advocate the use of random regression mixed models to estimate reaction-norm properties and the use of bivariate random regression to estimate selection on these properties within a single model. We illustrate the approach by random regression mixed models on 1115 observations of clutch sizes and laying dates of 361 female Ural owl Strix uralensis collected over 31 years to show that (1) there is variation across individuals in the slope of their clutch size-laying date relationship, and that (2) there is selection on the slope of the reaction norm between these two traits. Hence, natural selection potentially drives the negative covariance in clutch size and laying date in this species. The random-regression approach is hampered by inability to estimate nonlinear selection, but avoids a number of disadvantages (stats-on-stats, connecting reaction-norm properties to fitness). The approach is of value in describing and studying selection on behavioral reaction norms (behavioral syndromes) or life-history reaction norms. The approach can also be extended to consider the genetic underpinning of reaction-norm properties.     

Quantification of random mutations in the mitochondrial genome

Mitochondrial DNA (mtDNA) mutations contribute to the pathology of a number of age-related disorders, including Parkinson disease [A. Bender et al., Nat. Genet. 38 (2006) 515,Y. Kraytsberg et al., Nat. Genet. 38 (2006) 518], muscle-wasting [J. Wanagat, Z. Cao, P. Pathare, J.M. Aiken, FASEB J. 15 (2001) 322], and the metastatic potential of cancers [K. Ishikawa et al., Science 320 (2008) 661]. The impact of mitochondrial DNA mutations on a wide variety of human diseases has made it increasingly important to understand the mechanisms that drive mitochondrial mutagenesis. In order to provide new insight into the etiology and natural history of mtDNA mutations, we have developed an assay that can detect mitochondrial mutations in a variety of tissues and experimental settings [M. Vermulst et al., Nat. Genet. 40 (2008) 4, M. Vermulst et al., Nat. Genet. 39 (2007) 540]. This methodology, termed the Random Mutation Capture assay, relies on single-molecule amplification to detect rare mutations among millions of wild-type bases [J.H. Bielas, L.A. Loeb, Nat. Methods 2 (2005) 285], and can be used to analyze mitochondrial mutagenesis to a single base pair level in mammals.     

A stochastic model of cell aggregation under planar flow in the dilute regime

Models of the adhesion of a population of cells in a plane flow are developed, considering the dilute regime. Cells considered as rigid punctual entities are virtually injected at regular times within a plane channel limited by two fixed planes. The pressure profile is supposed to be triangular (constant gradient), in accordance with the assumptions of a Poiseuille flow. The cell adherence to the channel wall is governed by the balance of forces, accounting for gravity, non-specific physical interactions, such as electrostatic effects (repulsive) and Van der Waals forces (attractive), specific adhesive forces representing the ligand–receptor interactions, and friction between cells and the fluid in the vicinity of the endothelium wall. The spatial distribution of the adhesion molecules along the wall is supposed to be a random event, accounted for by a stochastic spatial variability of the dipolar moments of those molecules, according to a Gaussian process. Experimental trends reported for the rate of aggregation of L-selectin mediated leukocytes under shear flow are in qualitative accordance with the evolution versus time of adhering cells obtained by the present simulations. The effect of the maximal injection pressure on those kinetics is assessed.     

A stochastic model in tumor growth

A stochastic model of solid tumor growth based on deterministic Gompertz law is presented. Tumor cells evolution is described by a one-dimensional diffusion process limited by two absorbing boundaries representing healing threshold and patient death (carrying capacity), respectively. Via a numerical approach the first exit time problem is analysed for the process inside the region restricted by the boundaries. The proposed model is also implemented to simulate the effects of a time-dependent therapy. Finally, some numerical results are obtained for the specific case of a parathyroid tumor.     

Selection of cell specific peptides in a rat carotid injury model using a random peptide-presenting bacterial library

Cell specific peptides are possible candidates to enable targeted delivery of drugs and therapeutic genes in vivo. This study explores the utility of using a peptide-presenting bacterial library (pFliTrx) for the selection of new cell specific peptides, which bind to vascular cells of perfused tissues or organs. The balloon-injured rat carotid artery served as a model. Following perfusion of injured vascular segments with pFliTrx, 36 single clones could be identified. In radioligand binding studies, one of them, peptide P36, binds predominantly to perfused injured versus control vessel segments. It was additionally found that P36 binds with a 700-fold higher affinity in vitro to endothelial cells stimulated by treatment with LPS and TNF-alpha compared with unstimulated endothelial cells. The amino acid sequence of P36 reveals high homology to alpha(4)beta(1)-integrin, which mediates leukocyte migration from the vasculature at sites of inflammation via binding to cellular adhesion molecules, such as VCAM. In summary, this study demonstrates, that high specific peptides directed against injured vascular cells can be selected using a random peptide-presenting bacterial library.     

Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment

A new single-species model disturbed by both white noise and colored noise in a polluted environment is developed and analyzed. Sufficient criteria for extinction, stochastic nonpersistence in the mean, stochastic weak persistence in the mean, stochastic strong persistence in the mean and stochastic permanence of the species are established. The threshold between stochastic weak persistence in the mean and extinction is obtained. The results show that both white and colored environmental noises have sufficient effect to the survival results.     

Robustness of hen lysozyme monitored by random mutations

We investigated the robustness of hen lysozyme by using random mutant libraries. Six random mutant libraries containing 1, 1.5, 2, 3, 5 and 14 amino acid mutations per hen lysozyme were systematically constructed by varying the concentrations of Mg(2+) and Mn(2+) on polymerase chain reaction. The mutated genes from the six libraries were cloned to a yeast expression vector and a total of 4000 clones were screened on the basis of lysis activity and ELISA employing monoclonal antibody that recognized only lysozyme with native conformation. About 80% of the clones with an average of two amino acid mutations retained active structure. Almost all clones with an average of five mutations lost active structure. On the other hand, 80% of the clones with an average of two amino acid mutations retained both gross conformation and active structure and 24% of the clones with an average of 14 amino acid mutations retained gross conformation. These results show that gross conformation is robust against mutations and so is active structure to a lesser extent.     

Beyond the random coil: stochastic conformational switching in intrinsically disordered proteins

Intrinsically disordered proteins (IDPs) participate in critical cellular functions that exploit the flexibility and rapid conformational fluctuations of their native state. Limited information about the native state of IDPs can be gained by the averaging over many heterogeneous molecules that is unavoidable in ensemble approaches. We used single molecule fluorescence to characterize native state conformational dynamics in five synaptic proteins confirmed to be disordered by other techniques. For three of the proteins, SNAP-25, synaptobrevin and complexin, their conformational dynamics could be described with a simple semiflexible polymer model. Surprisingly, two proteins, neuroligin and the NMDAR-2B glutamate receptor, were observed to stochastically switch among distinct conformational states despite the fact that they appeared intrinsically disordered by other measures. The hop-like intramolecular diffusion found in these proteins is suggested to define a class of functionality previously unrecognized for IDPs.     

Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment II

This is a continuation of our paper [Liu, M., Wang, K., 2010. Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, J. Theor. Biol. 264, 934-944]. Taking both white noise and colored noise into account, a stochastic single-species model under regime switching in a polluted environment is studied. Sufficient conditions for extinction, stochastic nonpersistence in the mean, stochastic weak persistence and stochastic permanence are established. The threshold between stochastic weak persistence and extinction is obtained. The results show that a different type of noise has a different effect on the survival results.     

Selection for altruism through random drift in variable size populations

Background

Seed storage proteins are a major source of dietary protein, and the content of such proteins determines both the quantity and quality of crop yield. Significantly, examination of the protein content in the seeds of crop plants shows a distinct difference between monocots and dicots. Thus, it is expected that there are different evolutionary patterns in the genes underlying protein synthesis in the seeds of these two groups of plants.

Results

Gene duplication, evolutionary rate and positive selection of a major gene family of seed storage proteins (the 11S globulin genes), were compared in dicots and monocots. The results, obtained from five species in each group, show more gene duplications, a higher evolutionary rate and positive selections of this gene family in dicots, which are rich in 11S globulins, but not in the monocots.

Conclusion

Our findings provide evidence to support the suggestion that gene duplication and an accelerated evolutionary rate may be associated with higher protein synthesis in dicots as compared to monocots.     

Selection for altruism through random drift in variable size populations

ABSTRACT: BACKGROUND: Altruistic behavior is defined as helping others at a cost to oneself and a lowered fitness. The lower fitness implies that altruists should be selected against, which is in contradiction with their widespread presence is nature. Present models of selection for altruism (kin or multilevel) show that altruistic behaviors can have 'hidden' advantages if the 'common good' produced by altruists is restricted to some related or unrelated groups. These models are mostly deterministic, or assume a frequency dependent fitness. RESULTS: Evolutionary dynamics is a competition between deterministic selection pressure and stochastic events due to random sampling from one generation to the next. We show here that an altruistic allele extending the carrying capacity of the habitat can win by increasing the random drift of "selfish" alleles. In other terms, the fixation probability of altruistic genes can be higher than those of a selfish ones, even though altruists have a smaller fitness. Moreover when populations are geographically structured, the altruists advantage can be highly amplified and the fixation probability of selfish genes can tend toward zero. The above results are obtained both by numerical and analytical calculations. Analytical results are obtained in the limit of large populations. CONCLUSIONS: The theory we present does not involve kin or multilevel selection, but is based on the existence of random drift in variable size populations. The model is a generalization of the original Fisher-Wright and Moran models where the carrying capacity depends on the number of altruists.