Mutation of albedo and growth response produces oscillations in a spatial Daisyworld

We extended a two-dimensional cellular automaton (CA) Daisyworld to include mutation of optimal growth temperature as well as mutation of albedo. Thus, the organisms (daisies) can adapt to prevailing environmental conditions or evolve to alter their environment. We find the resulting system oscillates with a period of hundreds of daisy generations. Weaker and less regular oscillations exist in previous daisyworld models, but they become much stronger and more regular here with mutation in the growth response. Despite the existence of a particular combination of mean albedo and optimum individual growth temperature which maximises growth, we find that this global state is unstable with respect to mutations which lower absolute growth rate, but increase marginal growth rate. The resulting system oscillates with a period that is found to decrease with increasing death rate, and to increase with increasing heat diffusion and heat capacity. We speculate that the origin of this oscillation is a Hopf bifurcation, previously predicted in a zero-dimensional system.     

A novel mating system analysis for modes of self-oriented mating applied to diploid and polyploid arctic Easter daisies (Townsendia hookeri)

We have developed a new model for mating system analysis, which attempts to distinguish among alternative modes of self-oriented mating within populations. This model jointly estimates the rates of outcrossing, selfing, automixis and apomixis, through the use of information in the family structure given by dominant genetic marker data. The method is presented, its statistical properties evaluated, and is applied to three arctic Easter daisy populations, one consisting of diploids, the other two of tetraploids. The tetraploids are predominantly male sterile and reported to be apomictic while the diploids are male fertile. In each Easter daisy population, 10 maternal arrays of six progeny were assayed for amplified fragment length polymorphism markers. Estimates, confirmed with likelihood ratio tests of mating hypotheses, showed apomixis to be predominant in all populations (ca. 70%), but selfing or automixis was moderate (ca. 25%) in tetraploids. It was difficult to distinguish selfing from automixis, and simulations confirm that with even very large sample sizes, the estimates have a very strong negative statistical correlation, for example, they are not independent. No selfing or automixis was apparent in the diploid population, instead, moderate levels of outcrossing were detected (23%). Low but significant levels of outcrossing (2-4%) seemed to occur in the male-sterile tetraploid populations; this may be due to genotyping error of this level. Overall, this study shows apomixis can be partial, and provides evidence for higher levels of inbreeding in polyploids compared to diploids and for significant levels of apomixis in a diploid plant population.     

Dynamic membrane structure induces temporal pattern formation

The understanding of temporal pattern formation in biological systems is essential for insights into regulatory processes of cells. Concerning this problem, the present work introduces a model to explain the attachment/detachment cycle of MARCKS and PKC at the cell membrane, which is crucial for signal transduction processes. Our model is novel with regard to its driving mechanism: Structural changes within the membrane fuel an activator–inhibitor based global density oscillation of membrane related proteins. Based on simulated results of our model, phase diagrams were generated to illustrate the interplay of MARCKS and PKC. They predict the oscillatory behavior in the form of the number of peaks, the periodic time, and the damping constant depending on the amounts of MARCKS and PKC, respectively. The investigation of the phase space also revealed an unexpected intermediate state prior to the oscillations for high amounts of MARCKS in the system. The validation of the obtained results was carried out by stability analysis, which also accounts for further enhanced understanding of the studied system. It was shown, that the occurrence of the oscillating behavior is independent of the diffusion and the consumption of the reactants. The diffusion terms in the used reaction–diffusion equations only act as modulating terms and are not required for the oscillation. The hypothesis of our work suggests a new mechanism of temporal pattern formation in biological systems. This mechanism includes a classical activator–inhibitor system, but is based on the modifications of the membrane structure, rather than a reaction–diffusion system.     

A simulation study of the genetic regulatory hierarchy for butterfly eyespot focus determination

The color patterns on the wings of butterflies have been an important model system in evolutionary developmental biology. Two types of models have been used to study these patterns. The first type of model employs computational techniques and generalized mechanisms of pattern formation to make predictions about how color patterns will vary as parameters of the model are changed. These generalized mechanisms include diffusion gradient, reaction-diffusion, lateral inhibition, and threshold responses. The second type of model uses known genetic interactions from Drosophila melanogaster and patterns of candidate gene expression in one of several butterfly species (most often Junonia (Precis) coenia or Bicyclus anynana) to propose specific genetic regulatory hierarchies that appear to be involved in color pattern formation. This study combines these two approaches using computational techniques to test proposed genetic regulatory hierarchies for the determination of butterfly eyespot foci (also known as border ocelli foci). Two computer programs, STELLA 8.1 and Delphi 2.0, were used to simulate the determination of eyespot foci. Both programs revealed weaknesses in a genetic model previously proposed for eyespot focus determination. On the basis of these simulations, we propose two revised models for eyespot focus determination and identify components of the genetic regulatory hierarchy that are particularly sensitive to changes in model parameter values. These components may play a key role in the evolution of butterfly eyespots. Simulations like these may be useful tools for the study of other evolutionary developmental model systems and reveal similar sensitive components of the relevant genetic regulatory hierarchies.     

The regulatory capacity of Turing's model for morphogenesis, with application to slime moulds

For a one-dimensional system (elongated organism), regulation of a pattern of alternating regions of two cell types against disturbance by growth or or by cutting is discussed. Turing's linear two-morphogen model requires the number of regions to be that which maximizes the exponential growth rate of morphogen concentrations. On this basis, a diagram is constructed to show ranges of system length for patterns with various numbers of regions. Experimental features of pattern in the Dictyostelium slug, especiially the wide size range having regulatory capacity and the inequality of numbers of the two cell types, are compared with the simple model and with extensions of it to include dependence of diffusivity on cell type and morphogen transport by movement of whole cells.     

The effect of nest moisture on daily temperature regime in the nests of Formica polyctena wood ants

Summary: Relations between nest moisture and daily temperature regime were studied. Two extreme situations were distinguished among the variable patterns of daily temperature regime found. The first one was characterized by increasing temperature during the day and decreasing temperature during the night and was typical for dry nests exposed to sun. The second one was characterized by a lower temperature during the day that increased rapidly during the night. This pattern occurred in moist shaded nests and was less frequent than the first one.¶Patterns of surface nest temperature, which closely correspond with thermal loss, differ between dry and wet nests as well. In dry nests, the temperature decreased during the night on the whole surface in a similar way. In moist nests, the top parts constantly had a high temperature. The average surface temperature during the night was significantly higher in moist than in dry nests, which implies a higher thermal loss in moist nests. Microbial respiration of nest material strongly correlated with nest moisture, which implies a higher microbial heat production in moist nests.¶The results indicated two mechanisms for the maintenance of internal nest temperature. The first one, used in dry nests, is based on a combination of ant metabolic heat production and the isolation properties of a dry nest. The second one, used in moist nests, is based on the metabolic heat production provided by both the ants and the microorganisms in the nest material. These two strategies differ in the pattern of daily temperature fluctuation.     

An introduction to the mathematical structure of the Wright–Fisher model of population genetics

In this paper, we develop the mathematical structure of the Wright–Fisher model for evolution of the relative frequencies of two alleles at a diploid locus under random genetic drift in a population of fixed size in its simplest form, that is, without mutation or selection. We establish a new concept of a global solution for the diffusion approximation (Fokker–Planck equation), prove its existence and uniqueness and then show how one can easily derive all the essential properties of this random genetic drift process from our solution. Thus, our solution turns out to be superior to the local solution constructed by Kimura.     

Mesoscopic effects in an agent-based bargaining model in regular lattices

The effect of spatial structure has been proved very relevant in repeated games. In this work we propose an agent based model where a fixed finite population of tagged agents play iteratively the Nash demand game in a regular lattice. The model extends the multiagent bargaining model by Axtell, Epstein and Young modifying the assumption of global interaction. Each agent is endowed with a memory and plays the best reply against the opponent's most frequent demand. We focus our analysis on the transient dynamics of the system, studying by computer simulation the set of states in which the system spends a considerable fraction of the time. The results show that all the possible persistent regimes in the global interaction model can also be observed in this spatial version. We also find that the mesoscopic properties of the interaction networks that the spatial distribution induces in the model have a significant impact on the diffusion of strategies, and can lead to new persistent regimes different from those found in previous research. In particular, community structure in the intratype interaction networks may cause that communities reach different persistent regimes as a consequence of the hindering diffusion effect of fluctuating agents at their borders.     

Robustness of a gene regulatory circuit.

Complex interacting systems exhibit system behavior that is often not predictable from the properties of the component parts. We have tested a particular system property, that of robustness. The behavior of a system is termed robust if that behavior is qualitatively normal in the face of substantial changes to the system components. Here we test whether the behavior of the phage lambda gene regulatory circuitry is robust. This circuitry can exist in two alternative patterns of gene expression, and can switch from one regulatory state to the other. These states are stabilized by the action at the O(R) region of two regulatory proteins, CI and Cro, which bind with differential affinities to the O(R)1 and O(R)3 sites, such that each represses the synthesis of the other one. In this work, this pattern of binding was altered by making three mutant phages in which O(R)1 and O(R)3 were identical. These variants had the same qualitative in vivo patterns of gene expression as wild type. We conclude that the behavior of the lambda circuitry is highly robust. Based on these and other results, we propose a two-step pathway, in which robustness plays a key role, for evolution of complex regulatory circuitry.     

A mathematical theory of enamel solubility and the onset of dental caries: III. Development and computer simulation of a model of caries formation

Part III attempts to develop a diffusion controlled model of caries in the intact enamel employing the kinetic results of the previous two parts. A model of the enamel as a granular bed with a diffusible organic matrix filling the interstices is considered. The basic equations of diffusion and simultaneous reaction are developed under the assumption that all the reactions are so rapid as compared with the diffusion rate, that they are in a quasi-equilibrium state. The resultant system of seven coupled, non-linear parabolic partial differential equations is of such complexity that only numerical solutions could be attempted. Stability restrictions inherent in the problem dictated the use of the DuFort-Frankel numerical solution for parabolic boundary problems. Numerical solutions giving the concentration of all reactants, the rate of mineral loss, and the enamel porosity were obtained for a variety of boundary conditions. It is found that departure from the equilibrium condition expressed in part II is necessary for the occurrence of an attack on the enamel. The rate and pattern of penetration is then determined primarily by the concentrations of undissociated buffer, and salts, together with the rate of diffusion in the surrounding medium. The possibility of a relatively intact surface layer persisting over a demineralized subsurface region due solely to the composition of the demineralizing medium is noted. Remineralization behavior in portions of the carious lesion occurs in the model under certain boundary conditions.     

An Efficient, Nonlinear Stability Analysis for Detecting Pattern Formation in Reaction Diffusion Systems

Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I present a relatively simple and efficient, nonlinear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and nonlinear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have nonlinear patterning regimes not previously detected. The real power of this technique, however, is its simplicity and applicability to larger complex systems where other nonlinear methods become intractable. This is demonstrated through analysis of a chemotaxis regulatory network comprised of interacting proteins and phospholipids. In each case, predictions of this method are verified against results of numerical simulation, linear stability, asymptotic, and/or full PDE bifurcation analyses.     

Chemically Based Mathematical Model for Development of Cerebral Cortical Folding Patterns

The mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions, and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemically driven scenario, where an increase in intermediate progenitor cells in the subventricular zone correlates to gyral formation. Here we present a mathematical model that uses features of the IP model and further captures global characteristics of cortical pattern formation. A prolate spheroidal surface is used to approximate the ventricular zone. Prolate spheroidal harmonics are applied to a Turing reaction-diffusion system, providing a chemically based framework for cortical folding. Our model reveals a direct correlation between pattern formation and the size and shape of the lateral ventricle. Additionally, placement and directionality of sulci and the relationship between domain scaling and cortical pattern elaboration are explained. The significance of this model is that it elucidates the consistency of cortical patterns among individuals within a species and addresses inter-species variability based on global characteristics and provides a critical piece to the puzzle of cortical pattern formation.     

Numerical modelling of conductive and convective heat transfers in retinal laser applications

The control of the temperature increase is an important issue in retinal laser treatments. Within the fundus of the eye heat, generated by absorption of light, is transmitted by diffusion in the retinal pigment epithelium and in the choroid and lost by convection due to the choroidal blood flow. The temperature can be spatially and temporally determined by solving the heat equation. In a former analytical model this was achieved by assuming uniform convection for the whole fundus of the eye. A numerical method avoiding this unrealistic assumption by considering convective heat transfer only in the choroid is used here to solve the heat equation. Numerical results are compared with experimental results obtained by using a novel method of noninvasive optoacoustic retinal temperature measurements in rabbits. Assuming global convection the perfusion coefficient was evaluated to 0.07 s^?1, whereas a value of 0.32 s^?1 – much closer to values found in the literature (between 0.28 and 0.30 s^?1) – was obtained when choroidal convection was assumed, showing the advantage of the numerical method. The modelling of retinal laser treatment is thus improved and could be considered in the future to optimize treatments by calculating retinal temperature increases under various tissues and laser properties. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Surface topology assisted alignment of Min protein waves

Self-organization of proteins into large-scale structures is of pivotal importance for the organization of cells. The Min protein system of the bacterium Escherichia coli is a prime example of how pattern formation occurs via reaction–diffusion. We have previously demonstrated how Min protein patterns are influenced by compartment geometry. Here we probe the influence of membrane surface topology, as an additional regulatory element. Using microstructured membrane-clad soft polymer substrates, Min protein patterns can be aligned. We demonstrate that Min pattern alignment starts early during pattern formation and show that macroscopic millimeter-sized areas of protein patterns of well-defined orientation can be generated.     

Chaotic gene regulatory networks can be robust against mutations and noise

Robustness to mutations and noise has been shown to evolve through stabilizing selection for optimal phenotypes in model gene regulatory networks. The ability to evolve robust mutants is known to depend on the network architecture. How do the dynamical properties and state-space structures of networks with high and low robustness differ? Does selection operate on the global dynamical behavior of the networks? What kind of state-space structures are favored by selection? We provide damage propagation analysis and an extensive statistical analysis of state spaces of these model networks to show that the change in their dynamical properties due to stabilizing selection for optimal phenotypes is minor. Most notably, the networks that are most robust to both mutations and noise are highly chaotic. Certain properties of chaotic networks, such as being able to produce large attractor basins, can be useful for maintaining a stable gene-expression pattern. Our findings indicate that conventional measures of stability, such as damage propagation, do not provide much information about robustness to mutations or noise in model gene regulatory networks.