Non-equilibrium thermodynamics of the selection of biological macromolecules

The Eigen's model for the Darwinian selection of self-reproductive macromolecules is viewed by non-equilibrium thermodynamics of chemical reactions. An excess entropy suitable for the purpose is introduced. It, as the case may be, becomes a Liapunov function to investigate the stability of the selected state characterized by an extremum principle. The selection under the condition for the total concentration of self-reproductive macromolecular species to remain constant in time is discussed on the three cases; the selection described by the simplest Eigen's equation that the effect of mutation is neglected, the selection of quasi-species, and the selective behavior of elementary hypercycle.     

Stochastic Penna model for biological aging

Summary A stochastic genetic model for biological aging is introduced bridging the gap between the bit-string Penna model and the Pletcher-Neuhauser approach. The phenomenon of exponentially increasing mortality function at intermediate ages and its deceleration at advanced ages is reproduced for both the evolutionary steady-state population and the genetically homogeneous individuals.     

Some models for selection of biological macromolecules with time varying constraints

We consider some models for selection in self-reproducing macromolecular systems subject to time varying environmental constraints. We show that many of the results concerning selection obtained previously for stationary constraints may be generalized and that the over-all selective behaviour even in complex enzyme coupled systems is basically the same under the time varying external conditions as it is in the stationary case.     

Cryo-electron microscopy for structural analysis of dynamic biological macromolecules

Background

Since the introduction of what became today's standard for cryo-embedding of biological macromolecules at native conditions more than 30 years ago, techniques and equipment have been drastically improved and the structure of biomolecules can now be studied at near atomic resolution by cryo-electron microscopy (cryo-EM) while capturing multiple dynamic states. Here we review the recent progress in cryo-EM for structural studies of dynamic biological macromolecules.

The "sticky chain": a kinetic model for the deformation of biological macromolecules

The deformation behavior of certain biologic macromolecules is modeled by the "sticky chain," a freely jointed chain with weak bonds between subsequent joints. Straining the chain leads to thermally assisted breaking of the weak bonds, yielding a characteristic shape of the force-elongation curve, usually with a pronounced plateau, but sometimes displaying a pseudo-Hookean behavior over a wide range of deformations. The number of individual links is assumed to be large, so the stochastic time evolution of the individual events can be approximated by a differential equation. The cases of individual and collective bond breaking are treated and formulae given for various measurable quantities. A threshold strain rate is found, below which the deformation force no longer depends on the deformation velocity. The method is applied to experimental results for the deformation of single molecules like titin or DNA and the results agree with the parameters deduced from the same experiments by the original authors using Monte Carlo (MC) calculations. Despite its intrinsic continuous character, the model, therefore, is applicable even for the deformation of macromolecules with only a few discrete unfolding elements, yielding physical quantities from experimental results using simple formulae instead of a host of MC computations.     

The program XEASY for computer-supported NMR spectral analysis of biological macromolecules

Summary A new program package, XEASY, was written for interactive computer support of the analysis of NMR spectra for three-dimensional structure determination of biological macromolecules. XEASY was developed for work with 2D, 3D and 4D NMR data sets. It includes all the functions performed by the precursor program EASY, which was designed for the analysis of 2D NMR spectra, i.e., peak picking and support of sequence-specific resonance assignments, cross-peak assignments, cross-peak integration and rate constant determination for dynamic processes. Since the program utilizes the X-window system and the Motif widget set, it is portable on a wide range of UNIX workstations. The design objective was to provide maximal computer support for the analysis of spectra, while providing the user with complete control over the final resonance assignments. Technically important features of XEASY are the use and flexible visual display of strips, i.e., two-dimensional spectral regions that contain the relevant parts of 3D or 4D NMR spectra, automated sorting routines to narrow down the selection of strips that need to be interactively considered in a particular assignment step, a protocol of resonance assignments that can be used for reliable bookkeeping, independent of the assignment strategy used, and capabilities for proper treatment of spectral folding and efficient transfer of resonance assignments between spectra of different types and different dimensionality, including projected, reduced-dimensionality triple-resonance experiments.Abbreviations 1D, 2D, 3D, 4D one-, two-, three-, four-dimensional - NOE nuclear Overhauser enhancement - NOESY nuclear Overhauser enhancement spectroscopy - TOCSY total correlation spectroscopy - COSY correlation spectroscopy - TPPI time-proportional phase incrementation     

Stability analysis of a stochastic model for biomolecular selection

The technique of the probability generating function is used to derive the stochastic differential equations for a nonlinear model based on Eigen and Schuster's theory of biomolecular selection and evolution. The stabilities of various steady states are analyzed by using the linear stability approximation. The instability of a small starting population is investigated numerically. The minimum starting populations required for steady-state survival are then estimated for a wide range of parameters.     

Stochastic model of domain kinetics in biological membranes

A stochastic storage model based on the behavior of macroscopic variables of the system is used to describe the kinetics of raft-like domains in biological membranes. For a simple output model, we examine the features of the system behavior corresponding to the noise-induced nonequilibrium phase transitions. Characteristics of the behavior of the statistical system are obtained: an explicit form of the stationary distribution of the number of domains; ratios for the phase transition points; the expression for the first two moments of the random domain concentration, and the expression for the lifetime of membrane domains in the stationary state.     

Microcalorimetry of biological macromolecules

The capabilities of contemporary differential scanning and isothermal titration microcalorimetry for studying the thermodynamics of protein unfolding/refolding and their association with partners, particularly target DNA duplexes, are considered. It is shown that the predenaturational changes of proteins must not be ignored in studying the thermodynamics of formation of their native structure and their complexes with partners, particularly their cognate DNA duplexes.     

Global positive coexistence of a nonlinear elliptic biological interacting model

The purpose of this note is to give a necessary and sufficient condition for the coexistence of positive solutions to a rather general type of elliptic predator-prey system of the Dirichlet problem on the bounded domain omega when omega is a subset of Rn is large. The result is that the partial differential equation system possesses positive coexistence if and only if the corresponding ordinary differential equation system has positive equilibrium, the positive constant states. This result thus yields an algebraically computable criterion for the positive coexistence of predator and prey in many biological models.     

Stochastic theory of a selection game

The stochastic theory of a nonlinear game is presented which incorporates some of the essential properties of living systems: metabolism, reproduction and mutability. The steady state distribution function as well as the complete time development are given explicitly. The second law of thermodynamics is generalized to a certain class of nonequilibrium systems. An order parameter is introduced as a measure of the system's internal organization. From the point of view of phase transition theory, the model exhibits a transition at the absolute zero of temperature, with critical behaviour showing up in the low temperature region.     

Stochastic finite element analysis of biological systems: comparison of a simple intervertebral disc model with experimental results

Statistical methods allow the effects of uncertainty to be incorporated into finite element models. This has potential benefits for the analysis of biological systems where natural variability can give rise to substantial uncertainty in both material and geometrical properties. In this study, a simple model of the intervertebral disc under compression was created and analysed as both a deterministic and a stochastic system. Factorial analysis was used to determine the important parameters to be included in the stochastic analysis. The predictions from the model were compared to experimental results from 21 sheep discs. The size and shape of the distribution of the axial deformations predicted by the model was consistent with the experimental results given that the number of model solutions far exceeded the number of experimental results. Stochastic models could be valuable in determining the range and most likely value of stress in a tissue or implant.     

First and second moments and the mean hamming distance in a stochastic replication-mutation model for biological macromolecules

In this work first and second moments for a many species Moran model are calculated. The model describes by means of a time-continuous birth- and death process the evolution of an ensemble of N macromolecules out of n possible species. The molecules may replicate (correct or erroneous, in the latter case producing mutants) and may undergo elimination. Replication and elimination will be coupled in order to keep population size constant. In the case of arbitrary replication rates an expansion of the moments in powers of 1/N is found. For equal replication rates exact calculation of the moments is possible. In the case of a v-cube model (binary macromolecules) the second moments may be used to find a simple expression for the mean Hamming distance in the system. This quantity provides a measure for the localization of the ensemble.Supported by Stiftung Volkswagenwerk     

Weakly nonlinear analysis of a hyperbolic model for animal group formation

We consider an one-dimensional nonlocal hyperbolic model for group formation with application to self-organizing collectives of animals in homogeneous environments. Previous studies have shown that this model displays at least four complex spatial and spatiotemporal group patterns. Here, we use weakly nonlinear analysis to better understand the mechanisms involved in the formation of two of these patterns, namely stationary pulses and traveling trains. We show that both patterns arise through subcritical bifurcations from spatially homogeneous steady states. We then use these results to investigate the effect of two social interactions (attraction and alignment) on the structure of stationary and moving animal groups. While attraction makes the groups more compact, alignment has a dual effect, depending on whether the groups are stationary or moving. More precisely, increasing alignment makes the stationary groups compact, and the moving groups more elongated. Also, the results show the existence of a threshold for the total group density, above which, coordinated behaviors described by stationary and moving groups persist for a long time.      

The cybernetics of biological macromolecules

This paper develops the concept of linkage as it applies to the binding, of ligands by a polyfunctional macromolecule, or system of macromolecules, under equilibrium, steady-state, and transient conditions.     

Self-assembly of biological macromolecules.

The genetic apparatus of the cell is responsible for the accurate biosynthesis of the primary structure of macromolecules which then spontaneously fold up and, in certain circumstances, aggregate to yield the complex tertiary and quaternary structures of the biologically active molecules. Structures capable of self-assembly in this range from simple monomers through oligomers to complex multimeric structures that may contain more than one type of polypeptide chain and components other than protein. It is becoming clear that even with the simpler monomeric enzymes there is becoming clear that even with the simpler monomeric enzymes there is a kinetically determined pathway for the folding process and that a folded protein must now be regarded as the minimum free energy form of the kinetically accessible conformations. It is argued that the denatured subunits of oligomeric enzymes are likely to fold to something like their final structure before aggregating to give the native quaternary structure and the available evidence would suggest that this is so. The importance of nucleation events and stable intermediates in the self-assembly of more complex structures is clear. Many self-assembling structures contain only identical subunits and symmetry arguments are very successful in accounting for the structures formed. Because proteins are themselves complex molecules and not inelastic geometric objects, the rules of strict symmetry can be bent and quasi-equivalent bonding between subunits permitted. This possibility is frequently employed in biological structures. Conversely, symmetry arguments can offer a reliable means of choosing between alternative models for a given structure. It can be seen that proteins gain stability by growing larger and it is argued in evolutionary terms that aggregation of subunits is the preferred way to increase the size of proteins. The possession of quaternary structure by enzymes allows conferral of other biologically important properties, such as cooperativity between active sites, changes of specificity, substrate channelling and sequential reactions within a multi-enzyme complex. Comparison is made of the invariant subunit compositions of the simpler oligomeric enzymes with the variation evidently open to, say, the 2-oxoacid dehydrogenase complexes of E. coli. With viruses, on the other hand, the function of the quaternary structure is to package nucleic acid and, as an example, the assembly and breakdown of tobacco mosaic virus is discussed. Attention is drawn to the possible ways in which the principles of self-assembly can be extended to make structures more complicated than those that can be formed by simple aggregation of the comonent parts.     

A cooperative model for ligand binding to biological macromolecules as applied to oxygen carriers

This paper presents a model describing the thermodynamics of cooperative ligand binding to multimeric biological macromolecules and integrating some of the features of the two-state and induced-fit models. The protein is taken to be partitioned into a number of noninteracting functional constellations, each one existing in two possible quaternary conformations. Furthermore, the model postulates that a functional constellation is organized in several subsets of sites (called cooperons), in which subunits interact according to an induced-fit mechanism. In the present version the number of subunits forming a cooperon has been limited to two and the total number of parameters used for fitting experimental data is four, all having a precise physical meaning. Although the present application is limited to oxygen-carrying proteins (hemoglobins, hemocyanins, erythrocruorins), the model appears suitable to describe other biological macromolecules with functional interactions.     

Stochastic fluctuations in frequency-dependent selection: a one-locus, two-allele and two-phenotype model

Stochastic fluctuations in a simple frequency-dependent selection model with one-locus, two-alleles and two-phenotypes are investigated. The steady-state statistics of allele frequencies for an interior stable phenotypic equilibrium are shown to be similar to the stochastic fluctuations in standard evolutionary game dynamics [Tao, Y., Cressman, R., 2007. Stochastic fluctuations through intrinsic noise in evolutionary game dynamics. Bull. Math. Biol. 69, 1377-1399]. On the other hand, for an interior stable phenotypic or genotypic equilibrium, our main results show that the deterministic model cannot be used to predict the expectation of phenotypic frequency. The variance of phenotypic frequency for an interior stable genotypic equilibrium is more sensitive to the expected population size than for an interior stable phenotypic equilibrium. Furthermore, the stochastic fluctuations of allele frequency and phenotypic frequency can be considered approximately independent of each other for these genotypic equilibria, but not for phenotypic.     

Bifurcation analysis of a nonlinear field model of regeneration

The analysis of bifurcating solutions in the Totafurno and Trainor [23] model of supernumerary limb production in salamanders is re-examined using the symmetry analysis developed by Totafurno [22]. In particular, we show analytically that the appearance of field solutions possessing 2 and 4 singularities (the 2- and 4-centered solutions, respectively) also correspond to true bifurcations with reduced symmetries, just as had been previously found for a solution to the field equations not possessing such singularities (the twist solution). While the results have significance primarily for the biological problem, this work serves as an instructive example of the application of symmetry groups to the bifurcation analysis of nonlinear field equations arising from a variational principle. The relationship between the solutions of the nonlinear equations and the corresponding linear equations is discussed.Supported by the National Sciences and Engineering Research Council and the Medical Research Council of CanadaTo whom correspondence should be sent