Testing a model for the dynamics of actin structures with biological parameter values

A simple mathematical model for the dynamics of network-bundle transitions in actin filaments has been previously proposed and some of its mathematical properties have been described. Other models in this class have since been considered and investigated mathematically. In this paper, we have made the first steps in connecting parameters in the model with biologically measurable quantities such as published values of rate constants for filament-crosslinker association. We describe how this connection was made, and give some preliminary numerical simulation results for the behavior of the model under biologically realistic parameter regimes. A key result is that filament length influences the bundle-network transition.     

Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems

The need to study spatio-temporal chaos in a spatially extended dynamical system which exhibits not only irregular, initial-value sensitive temporal behavior but also the formation of irregular spatial patterns, has increasingly been recognized in biological science. While the temporal aspect of chaotic dynamics is usually characterized by the dominant Lyapunov exponent, the spatial aspect can be quantified by the correlation length. In this paper, using the diffusion-reaction model of population dynamics and considering the conditions of the system stability with respect to small heterogeneous perturbations, we derive an analytical formula for an ‘intrinsic length’ which appears to be in a very good agreement with the value of the correlation length of the system. Using this formula and numerical simulations, we analyze the dependence of the correlation length on the system parameters. We show that our findings may lead to a new understanding of some well-known experimental and field data as well as affect the choice of an adequate model of chaotic dynamics in biological and chemical systems.     

Relative embryo length as an adaptation to habitat and life cycle in Apiaceae

? The factors driving the evolution of the relative embryo length in Apiaceae were examined. We tested the hypothesis that seeds with large relative embryo length, because of more rapid germination, are beneficial in dry and open habitats and for short-lived species. We also analyzed to what extent delayed germination as a result of embryo growth can be considered a dormancy mechanism. ? Hypotheses were tested by correlating the relative embryo length with other plant traits, habitat and climatic variables. The adaptive nature of the relative embryo length was determined by comparing the performance of a pure drift, Brownian motion (BM) model of trait evolution with that of a selection-inertia, Ornstein-Uhlenbeck (OU) model. ? A positive correlation of the relative embryo length with germination speed and negative correlations with the amount of habitat shade, longevity and precipitation were found. An OU model, in which the evolution of longer embryos corresponded to a transition to habitats of high light, or to a short life cycle, outperformed significantly a BM model. ? The results indicated that the relative embryo length may have evolved as an adaptation to habitat and life cycle, whereas dormancy was mainly related to temperature at the sampling sites.     

Actin cross-linkers and the shape of stereocilia

Stereocilia are actin-based cellular protrusions essential for hearing. We propose that they are shaped by the detachment dynamics of actin cross-linkers, in particular espin. We account for experimentally observed stereocilium shapes, treadmilling velocity to length relationship, espin 1 localization profile, and microvillus length to espin level relationship. If the cross-linkers are allowed to reattach, our model yields a dynamical phase transition toward unbounded growth. Considering the simplified case of a noninteracting, one-filament system, we calculate the length probability distribution in the growing phase and its stationary form in a continuum approximation of the finite-length phase. Numerical simulations of interacting filaments suggest an anomalous power-law divergence of the protrusion length at the growth transition, which could be a universal feature of cross-linked depolymerizing systems.     

Evaluation and Estimation of Various Markov Models with Applications to Membrane Channel Kinetics

Hidden Markov modelling is a powerful and efficient digital signal processing strategy for extracting the maximum likelihood model from a finite length sample of noisy data. Assuming the number of states in the model is known, then the state levels, transition probabilities, initial state distribution and the noise variance can be estimated. We investigate the applicability of this technique in membrane channel kinetics not only as a parameter estimator, but also as an aid to discriminating between various model types according to their statistical likelihood. We survey three representative classes of channel dynamics, namely: aggregated Markov models, semi-Markov models (with asymptotically convergent transition probabilities), and coupled Markov models; reformulating each within a discrete-time hidden Markov model framework. We then provide numerical evidence of the effectiveness of the procedure using simulated channel data and hence show that the correct model, as well as the model parameters, can be discerned. We also demonstrate that the model likelihood can be used to indicate the approximate number of states in the model.     

On the botanic model of plant growth with intermediate vegetative-reproductive stage

The application of dynamic optimization to mathematical models of ontogenic biological growth has been the subject of much research [see e.g. . J. Theor. Biol. 33, 299-307]. Koz?owsky and Zió?ko [1988. Thor. Popul. Biol. 34, 118-129] and Zió?ko and Koz?owski [1995. IEEE Trans. Automat. Contr. 40(10), 1779-1783] presented a model with gradual transition from vegetative to reproductive growth. The central point of their model is a mixed state-control constraint on the rate of reproductive growth, which leads to a mixed vegetative-reproductive growth period. Their model is modified here in order to take into account the difference of photosynthesis use efficiency when energy is accumulated in the vegetative and in the reproductive organs of a plant, respectively. The simple assumption on correlation between photosynthesis and temperature permits us to modify the model in a form that is useful for changing climate. Unfortunately, the mathematical solution of the optimal control problem in Koz?owsky and Zió?ko (1988) and Zió?ko and Koz?owski (1995) is incorrect. The strict mathematical solution is presented here, the numerical example from is solved, and the results are compared. The influence of the length of the season and the relative photosynthesis use efficiency, as well as of the potential sink demand of the reproductive organs, on the location and duration of the mixed vegetative-reproduction period of growth is investigated numerically. The results show that the mixed growth period is increased and shifted toward the end of the season when the lengths of the season is increased. Additional details of the sensitivity analysis are also presented.     

Phase transition in vector spin glasses

We first give an experimental and theoretical introduction to spin glasses, and then discuss the nature of the phase transition in spin glasses with vector spins. Results of Monte Carlo simulations of the Heisenberg spin glass model in three dimensions are presented. A finite size scaling analysis of the correlation length of the spins and chiralities shows that there is a single, finite-temperature transition at which both spins and chiralities order.     

Directional asymmetry in the limbs,skull and pelvis of the silver fox (V. vulpes)

Directional asymmetry (DA) is a characteristic of most vertebrates, most strikingly exhibited by the placement of various organs (heart, lungs, liver, etc.) but also noted in small differences in the metrics of skeletal structures such as the pelvis of certain fish or sauropsids. We have analyzed DA in the skeleton of the fox (V. vulpes), using ～1,000 radiographs of foxes from populations used in the genetic analysis of behavior and morphology. Careful measurements from this robust data base demonstrate that: 1) DA occurs in the limb bones, the ileum, and ischium and in the mandible; 2) regardless of the direction of the length asymmetry vector of a particular skeletal unit, the vectorial direction of length is always opposite to that of width; 3) with the exception of the humerus and radius, there is no correlation or inverse correlation between vectorial amplitudes or magnitudes of bone asymmetries. 4) Postnatal measurements on foxes demonstrate that the asymmetry increases after birth and continues to change (increasing or decreasing) during postnatal growth. 5) A behavior test for preferential use of a specific forelimb exhibited fluctuating asymmetry but not DA. None of the skeletal asymmetries were significantly correlated with a preferential use of a specific forelimb. We suggest that for the majority of fox skeletal parameters, growth on the right and left side of the fox are differentially biased resulting in fixed differences between the two sides in either the rate of growth or the length of the period during which growth occurs. Random effects around these fixed differences perturb the magnitude of the effects such that the magnitudes of length and width asymmetries are not inversely correlated at the level of individual animals. J. Morphol., 2010. © 2010 Wiley‐Liss, Inc.     

Applying the stochastic difference equation to DNA conformational transitions: a study of B-Z and B-A DNA transitions

Despite the existence of numerous models to account for the B-Z DNA transition, experimenters have not yet arrived at a conclusive answer to the structural and dynamical features of the B-Z transition. By applying the stochastic difference equation to simulate the B-Z DNA transition, we have shown that the stretched intermediate model of the B-Z transition is more probable than other B-Z transition models such as the Harvey model. This is accomplished by comparing potential energy profiles of various B-Z DNA transition models and calculating relative probabilities based on the stochastic difference equation with respect to length (SDEL) formalism. The results garnered in this article allow for new approaches in determining the structural transition of B-DNA to Z-DNA experimentally. We have also simulated the B-A DNA transition using the stochastic difference equation. Unlike the B-Z DNA transition, the mechanism for the B-A DNA transition is well established. The variation in the pseudorotation angle during the transition is in good agreement with experimental results. Qualitative features of the simulated B-A transition also agree well with experimental data. The SDEL approach is thus a suitable numerical technique to compute long-time molecular dynamics trajectory for DNA molecules.     

Scaling analysis of the concentration dependence on elasticity of agarose gel

Since the critical exponent of the elastic modulus is related to the spatial dimension and the critical exponent of the correlation length, depending on the characteristics of elasticity, we experimentally evaluated both the elastic modulus of a sol-gel transition system and also the correlation length. We could determine the correlation length of agarose gel by the dynamic light scattering method; it was well described by the power law as a function of the deviation from the sol-gel transition point. Three scaling laws between the critical exponent of the correlation length (v) and that of the elastic shear modulus (t) were compared, and the critical exponent of the elastic modulus was described by the equation of de Gennes expression (t=1+v(d-2), where d is the spatial dimension). This result suggests that agarose fibers are stiff enough to show scalar elasticity.     

A model for length control of flagellar hooks of Salmonella typhimurium

We present a mathematical model for the growth and length regulation of the hook component of the flagellar motor of Salmonella typhimurium. Under the assumption that the molecular constituents are translocated into the nascent filament by an ATP-ase and then move by molecular diffusion to the growing end, where they polymerize into the growing tube, we find that there is a detectable transition from secretion limited growth to diffusion limited growth. We propose that this transition can be detected by the secretant FliK, allowing FliK to interact with FlhB thereby changing the secretion target of the type III secretion machinery and terminating the growth of the hook.     

Model of Growth Cone Membrane Polarization via Microtubule Length Regulation

We present a mathematical model of membrane polarization in growth cones. We proceed by coupling an active transport model of cytosolic proteins along a two-dimensional microtubule (MT) network with a modified Dogterom-Leibler model of MT growth. In particular, we consider a Rac1-stathmin-MT pathway in which the growth and catastrophe rates of MTs are regulated by cytosolic stathmin, while the stathmin is regulated by Rac1 at the membrane. We use regular perturbation theory and numerical simulations to determine the steady-state stathmin concentration, the mean MT length distribution, and the resulting distribution of membrane-bound proteins. We thus show how a nonuniform Rac1 distribution on the membrane generates a polarized distribution of membrane proteins. The mean MT length distribution and hence the degree of membrane polarization are sensitive to the precise form of the Rac1 distribution and parameters such as the catastrophe-promoting constant and tubulin association rate. This is a consequence of the fact that the lateral diffusion of stathmin tends to weaken the effects of Rac1 on the distribution of mean MT lengths.     

Phase fluctuations on the micron-submicron scale in GUVs composed of a binary lipid mixture

We used a combination of imaging and fluctuation techniques to investigate the temporal evolution of gel phase domains at the onset of phase separation, as well as the correlation between domain topology and local lipid ordering in GUVs composed of a binary mixture of DPPC/DLPC 1:1. The data acquired at temperatures immediately above the transition temperature of the two lipids suggest fluctuations in the lipid organization with a lifetime <0.1 s and a characteristic length of 1.2 μm. As the temperature is decreased below the transition temperature of one of the lipids, coupling between the two leaflets of the bilayer is observed to begin within the first five minutes after the onset of phase separation. However, domains confined to only one leaflet can be found during the first 45-50 min after the onset of phase separation. Our analysis using a two-state model (liquid and gel) indicates that for the first 45-50 min from the onset of phase separation the two lipid phases do not strongly influence the phase behavior of each other on the micron-length scale. At longer times, behavior that deviates from the two-state model is observed and appears to be correlated to domain morphology.     

A model of the strain-induced B-Z transition

The B-to-Z transition in supercoiled circular DNA is modeled as a strain-induced nonlinear excitation process. Using a model, in which DNA is regarded as a chain of units with a bistable energy function along the twisting coordinate together with a harmonic inter-unit interaction, we show that a Z region and the accompanying two B-Z junctions of finite width appear naturally as a solution of nonlinear equations, when the strain exceeds a critical value. We examine the B-Z transition behaviour as a function of twist under various situations. We also analyse available experimental results on B-Z transition in supercoiled plasmid with G-C insertions by this mechanistic model in order to estimate the magnitude of model parameters. The energy barrier of the B-Z transition is estimated to be of the order of 1 kcal/mole per base pair. The analysis shows that if the length of the insertion is less than a certain value, the entire insertion converts to Z form at a transition point, but if the insertion is much longer, the B-Z transition exhibits a different behavior, in which part of the insertion flips to Z form and the Z region expands linearly upon changing linking number.     

Thermally induced proliferation of pores in a model fluid membrane.

The growth of thermally induced pores in a two-dimensional model fluid membrane is investigated by Monte Carlo simulation. Holes appear in the membrane via an activated process, and their subsequent growth is controlled by an edge energy per unit length or line tension. The barrier height and line tension, together with a lateral tension, are the independent parameters of the model. In the resulting phase diagram, a rupture transition separates an intact membrane from a disintegrated state. The approach to the ruptured state shows distinct regimes. Reducing the barrier height at large line tension produces multiple, quasi-independent, small holes whose behavior is dominated by their edge energy, whereas at lower line tensions shape fluctuations of the holes facilitate their coalescence into a single large hole. At a small value of line tension and large barrier height, a single hole spontaneously permeabilizes the membrane in an entropically driven phase transition. Entropy dominates pore growth for line tensions not far below those measured for artificial vesicles. Permeabilization of lipid bilayers by certain peptides involves perturbing lipid-lipid cohesive energies, and our simulations show that at small line tensions the entropy of hole shape fluctuations destroys the model membrane's stability.     

Helix-coil transition in heterogeneous DNA.

The broadening of a helix–coil transition due to base pair heterogeneity is calculated on the basis of a cumulant perturbation expansion in the quasi-grand ensemble. In this ensemble the fictitious, homogeneous chain, to which the perturbation is referred, automatically decreases its correlation length as the heterogeneity increases. This “renormalization” seems to stabilize the perturbation expansion, in view of the good agreement between the present results and the exact theory of a heterogeneous polypeptide helix–coil transition. For the DNA model in which ring entropy is included, the transitions is found to be extremely narrow for an infinite random chain with conventional parameters. A tentative reconciliation of this result with contradictory calculations of some other workers is offered on the basis of end effects, coarse graining, or approximation to the ring entropy. An application of the new method to DNA with a non-random base pair distribution requires evaluation of the correlation function between molecular states (helix or coil), at different sites of the reference chain. The evaluation is reduced to quadrature, but numerical calculations have been made only for the random chain.     

A Model of the Strain-induced B-Z Transition

Abstract

The B-to-Z transition in supercoiled circular DNA is modeled as a strain-induced nonlinear excitation process. Using a model, in which DNA is regarded as a chain of units with a bistable energy function along the twisting coordinate together with a harmonic inter-unit interaction, we show that a Z region and the accompanying two B-Z junctions of finite width appear naturally as a solution of nonlinear equations, when the strain exceeds a critical value. We examine the B-Z transition behaviour as a function of twist under various situations. We also analyse available experimental results on B-Z transition in supercoiled plasmid with G-C insertions by this mechanistic model in order to estimate the magnitude of model parameters. The energy barrier of the B-Z transition is estimated to be of the order of 1 kcal/mole per base pair. The analysis shows that if the length of the insertion is less than a certain value, the entire insertion converts to Z form at a transition point, but if the insertion is much longer, the B-Z transition exhibits a different behavior, in which part of the insertion flips to Z form and the Z region expands linearly upon changing linking number.