Statistical mechanical deconvolution of thermal transitions in macromolecules. III. Application to double-stranded to single-stranded transitions of nucleic acids

The statistical mechanical deconvolution theory for macromolecular conformational transitions is extended to the case of nucleic acids transitions involving strand separation. It is demonstrated that the partition function, Q, as well as all the relevant thermodynamic quantities of the system, can be calculated from experimental scanning calorimetric data. In particular, it is shown that important thermodynamic parameters such as the size of the average cooperative unit during strand separation, the mean helical segment length, and the mean coil-segment length can be calculated from the average excess enthalpy function 〈ΔH〉. The theory is applied to the double-stranded to single-stranded transition of the system poly(A)·poly(U) at different salt concentrations. It is shown that the mean helical segment length is a monotonically decreasing function of the temperature well before strand separation occurs. On the other hand, the mean coil segment length remains practically constant until temperatures very close to T_m. Both experimental findings clearly indicate that the unfolding of poly(A)·poly(U) proceeds through the formation of many short helical sequences. The cooperative unit for the strand separation is calculated to be about 120 base pairs and essentially independent of the salt concentration. The existence of a minimum helical segment length of 10 ± 2 base pairs within the double-stranded form is calculated.     

Estimation of molecular averages and equilibrium fluctuations in lipid bilayer systems from the excess heat capacity function

It is demonstrated that the bilayer partition function can be numerically obtained from scanning calorimetric data without assuming a particular model for the gel-liquid crystalline transition. From this partition function, the enthalpy, entropy and volume changes accompanying the transition can be calculated. In the limit of very large systems, the method of the grand partition function allows calculation of cluster model distribution functions from which average sizes of gel and liquid-crystal clusters, cluster densities and equilibrium fluctuations are obtained. These results indicate that the main transition in phospholipid bilayers proceeds through the formation of clusters and that these clusters are not static domains but highly fluctuating entities. These fluctuations in cluster size are approximately equal to the average cluster size and give rise to localized density and volume fluctuations. The magnitude of these fluctuations is affected by the radius of curvature of the bilayer and by the addition of small molecular weight compounds to the system.     

Theory of multivalent binding in one and two-dimensional lattices

Ligand binding to a linear lattice composed of N sites, under general conditions of cooperativity and number of sites covered upon binding, m, is approached in terms of the theory of contracted partition functions. The partition function of the system obeys a recursion relation leading to a generating function that provides an exact analytical solution for any case of interest. Site-specific properties of the lattice are derived from simple transformations of the analytical expressions. The McGhee-von Hippel model is obtained as a special case in the limit N --> infinity. The derivation is straightforward and involves no combinatorial arguments. Partition functions and site-specific properties are also derived for the case of non-cooperative binding to a two-dimensional torus of length N, containing s sites in its section for a total of sN sites. The torus provides a relevant model for ligand binding to double-stranded DNA (s = 2) or protein helices (s = 3,4). It is proved that non-cooperative binding to the two-dimensional torus can mimic cooperative binding to a one-dimensional linear lattice when m = s. The dimensional embedding of the lattice and the geometry of interaction of its sites play a crucial role in defining the binding properties of the system accessible to experimental measurements. Hence, caution must be exercised in the interpretation of Scatchard plots in terms of the one-dimensional McGhee-von Hippel model, especially when m < or = 4 and the geometry of the system is clearly two-dimensional.     

Statistical thermodynamics of oligomer-polymer interactions

A new method for formulating partition functions for a system of particles interacting on a one-dimensional lattice has recently been developed.^3,4 In this paper the method is applied to oligomer–polymer systems. The details of the connection between this method and the matrix formulation are first established. The method is then used to find expressions for the partition function of an oligomer–polymer system when the oligomers have a distribution of lengths and an alternating sequence.     

A rapid assay for the binding of actinomycin to DNA.

The theory of countercurrent distribution (CCD) was reviewed and extended. The separation function for the fundamental distribution of CCD was presented in the form n = t²(αk₁+β)²/αk₁(β?1)² where n is the number of transfers, t the abscissa of the standard normal distribution, α = v_m/v₈ the phase ratio, β = k₁/k₂≥ 1 the separation factor, and k₁ the partition coefficient of the more radidly moving component; n was found to be minimal on the condition αk₁ = β. The separation function for the single withdrawal of CCD was obtained in the form N = u + 1 = t²{(αk₁ + 1)^1/2 + [β(αk₁ + β)]^1/2}²/(β ? 1)²+ 1, where N is the number of partition units. From this equation it appears that N is minimal when αk₁ = 0. Compared with the former separation functions presented in the literature, these separation functions have the advantage of giving directly the relationships among the phase ratio, the absolute partition coefficient, the separation factor, the resolution degree, and the number of transfers or partition units required. In addition, the dependencies of the elution volumes and the widths of the elution curves on α, β, and the partition coefficients were considered mathematically by means of differential calculus. The elution volumes were found to have minima at certain αk₁ values. The standard deviations, on the contrary, did not have minima in respect to αk₁. The theory presented can be used for selecting proper operating conditions while separating chemical compounds.     

A grammatical theory for the conformational changes of simple helix bundles.

Polymers, including biomolecules such as proteins, have two particularly important types of single-molecule transitions: a helix-coil transition, driven by interactions that are local in the chain, and a collapse transition, driven by nonlocal interactions. A long-standing challenge of polymer statistical mechanics has been to deal with both types of transition in a single theoretical framework. The simplest paradigmatic problem would be a theory of helix-bundle folding. Here, we show how the machinery of formal grammars, originally developed in the context of linguistic analysis and programming-language compilation, provides a simple and general way to combine the Zimm-Bragg model of alpha-helices with the model of Chen and Dill for nonlocal interactions in antiparallel polymeric systems. We use a well-known construction in the theory of formal grammars to give the statistical mechanical partition function for two-helix bundles. Predictions are shown to be quite good in comparison to exact enumerations within a lattice model.     

Computing free energies using nested sampling-based approaches

Abstract

Nested sampling (NS) has emerged as a powerful statistical mechanical sampling technique to compute the partition function of atomic and molecular systems. From the partition function all thermodynamic quantities can be calculated in absolute terms, including absolute free energies and entropies. In this article, we provide a brief overview of NS within a Bayesian context, as well as overviews of how NS is used to compute the partition functions and thermodynamic quantities in the canonical and isothermal-isobaric ensembles. Then we introduce a new scheme, Coupling Parameter Path Nested Sampling, to estimate the free energy difference between two systems with different potential energy functions. The method uses a NS simulation to traverse the same path through phase space as would be covered in traditional coupling parameter-based methods such as thermodynamic integration and perturbation approaches. We demonstrate the new method with two case studies and confirm its accuracy by comparison to conventional methods, including Widom test particle insertion and thermodynamic integration. The proposed method provides a powerful alternative to traditional coupling parameter-based free energy simulation methods.     

构象电子系统的集体激发,跃迁过程和合作现象

本文给出了一套对分子生物学问题进行理论分析的方法并列举简单应用的实例.以分子构象和前沿电子为变数,引入了构家电子场,给出了研究其集体激发的Green函数途径,指出生物凝聚态中存在新型局域激发.导出了构象电子跃迁哈密顿量,特别研究了光致构家电子跃迁,指出低阶跃迁为非Franck—Condon型的.研究了链式分子的合作现象,阐明了振动激发对于实现合作转变的重要性及转变中可能存在反常温度依赖.     

Opto-thermal technologies for microscopic analysis of cellular temperature-sensing systems

Could enzymatic activities and their cooperative functions act as cellular temperature-sensing systems? This review introduces recent opto-thermal technologies for microscopic analyses of various types of cellular temperature-sensing system. Optical microheating technologies have been developed for local and rapid temperature manipulations at the cellular level. Advanced luminescent thermometers visualize the dynamics of cellular local temperature in space and time during microheating. An optical heater and thermometer can be combined into one smart nanomaterial that demonstrates hybrid function. These technologies have revealed a variety of cellular responses to spatial and temporal changes in temperature. Spatial temperature gradients cause asymmetric deformations during mitosis and neurite outgrowth. Rapid changes in temperature causes imbalance of intracellular Ca²⁺ homeostasis and membrane potential. Among those responses, heat-induced muscle contractions are highlighted. It is also demonstrated that the short-term heating hyperactivates molecular motors to exceed their maximal activities at optimal temperatures. We discuss future prospects for opto-thermal manipulation of cellular functions and contributions to obtain a deeper understanding of the mechanisms of cellular temperature-sensing systems.     

Parameterization of Coulomb interaction in three-dimensional periodic systems

A simple method is proposed to calculate Coulomb interactions in three-dimensional periodic cubic systems. It is based on the parameterization of the interaction on polynomials and rational functions. The parameterized functions are compared to tabulation methods, to the Ewald calculations and cubic harmonic function fits found in the literature. Our parameterizations are computationally more efficient than, the use of tabulations at all cases and seem to be more efficient than the cubic harmonic parameterizations in the case of simultaneous potential energy and force calculations. In comparison to the Ewald method, it is feasible to use the parameterizations on small systems and on systems, where pair-wise additive short-range interactions are dominant. One also may prefer the parameterizations to the Ewald method for large systems, if limited accuracy is needed. The embedding of the method into existing molecular dynamics and Monte Carlo simulation codes is very simple. The presented investigation contains some numerical experimental data to support the correct theoretical partition of potential energy in periodic systems, as well.     

Cooperative transition between two helical conformations in a linear system: Poly-L-proline I ⇌ II. I. Equilibrium studies

The solvent-induced conformational transition between the two helical forms of poly-L -proline is studied as a model for cooperative order ? order transitions. The chain length dependent equilibrium data in two solvent systems are described by Schwarz's theory, which is based upon the most general formulation of the linear Ising model with nearest neighbor interactions. The parameter σ which describes the difficulty of nucleation of a I (II) residue in an uninterrupted II (I)-helix is 10^?5 in both solvent systems. The ratios of the nucleation difficulties of states I and II at the ends of the chains β′ and β″ are very different in the two systems. Nucleation difficulty within the chain is interpreted as being due to unfavorable excess interaction energies at the I–II and II–I junctions, which add up to 7 kcal/mole of nuclei as calculated from the σ value. A similar value is computed from the atomic interactions at the junctions. In contrast to this intrinsic properly of poly-L -proline, the energies of I and II residues at the ends are heavily influenced by interactions of the endgroups with the solvent. The above values of the nucleation parameters are determined by a new least-square fitting procedure which does not necessitate the assumption of the dependence of the equilibrium constant s for propagation upon the external parameters, but yields this function from the experimental transition data. A quantitative explanation of this experimental s function through the binding of solvent is attempted. In the transition region a very small free energy change (about 0.1 kcal/mole), arising from a preferential binding of solvent molecules to one of the conformational states, is sufficient for a complete conversion from one helical form to the other.     

Confounding social and mating systems predictably lead to biased results when examining the evolution of cooperative breeding in cichlids: A response to Tanaka et al.

In 2017, we demonstrated that transitions to cooperative breeding in Lamprologine cichlid fishes were not related to a species’ social mating system (Dey et al. 2017. Nature Ecology & Evolution, 1 , 137). This contrasted previous evidence that monogamy (and a low degree of promiscuity) promoted transitions to cooperative breeding in other taxa. Recently, Tanaka et al. (2018. Ethology, 124 , 777–789) critiqued our study and argued that a re‐analysis of the data shows transitions to cooperative breeding are promoted by non‐monogamous mating systems. Here, we show that Tanaka et al.'s critique contains numerous inaccuracies. In addition, we show that the results put forth by Tanaka et al. emerge only under the extreme scenario in which all cooperative breeding species are classified as non‐monogamous, which we argue arises because Tanaka et al. confound social systems and mating systems. While we agree that there is uncertainty regarding the mating system of some Lamprologine species, we argue this uncertainty was sufficiently addressed through the extensive sensitivity analyses conducted in our original study.     

Ligand binding on ladder lattices

The ligand binding problems on two-dimensional ladders, which model many important binding phenomena in molecular biology, are studied in details. The model is represented by four parameters, the interactions between ligands when bound to adjacent sites on opposite legs of the ladder (tau), the interactions between bound ligands in the longitudinal direction of the ladder (sigma), the number of binding sites that are covered by a bound ligand (m), and the intrinsic binding constant (K). The partition functions of ring ladders are approached with the transfer matrix method. A general relation is derived which connects the partition function of a linear ladder with that of a ring ladder. The results obtained apply to the general situation of multivalent binding, in which m>1. Special attention is paid to the case where the ligand covers one site (m=1). In this case explicit formulas are given for the partition functions of ring and linear ladders. Closed-form expressions are obtained for various properties of the system, including the degree of binding (theta), the midpoint in the binding isotherm (1/square root(tau sigma)), the initial and end slopes of the Scatchard plots (2sigma + tau - 4 and -sigma2 tau, respectively). From these closed-form formulas, sigma and tau may be extracted from experimental data. The model reveals certain features which do not exist in one-dimensional models. Using the general method discussed in [1], the recurrence relation is found for the partition functions. The analytical solution found for this model provides test cases to verify the numerical results for more complex two-dimensional models.     

Geometric and probabilistic stability criteria for delay systems.

A new approach to the study of the stability of delay systems is developed. The method is applicable to biological control systems and other systems where little information about time delays is available. The view proposed is that stability information can be deduced from the statistical properties of the probability distribution that encodes the structure of the time delay. The main statistical variables used are the usual expectation parameter E and a modified variance, called relative variance and denoted R, that is invariant under time-scale changes. In many cases, the stability of a model improves as R increases while E remains fixed. The statistical approach is shown to be closely related to a geometric method of Walther and Cushing that establishes stability in the case of a convex delay distribution function. In fact, it is shown that convex and concave distributions have R values respectively greater than and less than 1/2. A generalized version of the geometric theory is presented that relaxes the smoothness hypothesis on the density function; this brings it more into correspondence with statistical theory, which applies to general distributions irrespective of their smoothness.     

Dynamic and coordinating domain motions in the active subunits of the F1-ATPase molecular motor

F1-ATPase is a rotary molecular motor crucial for various cellular functions. In F1-ATPase, the rotation of the gammadeltaepsilon subunits against the hexameric alpha(3)beta(3) subunits is highly coordinative, driven by ATP hydrolysis and structural changes at three beta subunits. However, the dynamical and coordinating structural transitions in the beta subunits are not fully understood at the molecular level. Here we examine structural transitions and domain motions in the active subunits of F1-ATPase via dynamical domain analysis of the alpha(3)beta(3)gammadeltaepsilon complex. The domain movement and hinge axes and bending residues have been identified and determined for various conformational changes of the beta-subunits. P-loop and the ATP-binding pocket are for the first time found to play essential mechanical functions additional to the catalytic roles. The cooperative conformational changes pertaining to the rotary mechanism of F1-ATPase appears to be more complex than Boyer's 'bi-site' activity. These findings provide unique molecular insights into dynamic and cooperative domain motions in F1-ATPase.     

Locally accessible conformations of proteins: multiple molecular dynamics simulations of crambin.

Multiple molecular dynamics (MD) simulations of crambin with different initial atomic velocities are used to sample conformations in the vicinity of the native structure. Individual trajectories of length up to 5 ns sample only a fraction of the conformational distribution generated by ten independent 120 ps trajectories at 300 K. The backbone atom conformational space distribution is analyzed using principal components analysis (PCA). Four different major conformational regions are found. In general, a trajectory samples only one region and few transitions between the regions are observed. Consequently, the averages of structural and dynamic properties over the ten trajectories differ significantly from those obtained from individual trajectories. The nature of the conformational sampling has important consequences for the utilization of MD simulations for a wide range of problems, such as comparisons with X-ray or NMR data. The overall average structure is significantly closer to the X-ray structure than any of the individual trajectory average structures. The high frequency (less than 10 ps) atomic fluctuations from the ten trajectories tend to be similar, but the lower frequency (100 ps) motions are different. To improve conformational sampling in molecular dynamics simulations of proteins, as in nucleic acids, multiple trajectories with different initial conditions should be used rather than a single long trajectory.     

Towards a calculus of biomolecular complexes at equilibrium

An overview is presented of the construction and use of algebraic partition functions to represent the equilibrium statistical mechanics of multimolecular complexes and their action within a larger regulatory network. Unlike many applications of equilibrium statistical mechanics, multimolecular complexes may operate with various subsets of their components present and connected to the others, the rest remaining in solution. Thus they are variable-structure systems. This aspect of their behavior may be accounted for by the use of 'fugacity' variables as a representation within the partition functions. Four principles are proposed by which the combinatorics of molecular complex construction can be reflected in the construction of their partition functions. The corresponding algebraic operations on partition functions are multiplication, addition, function composition and a less commonly used operation called contraction. Each has a natural interpretation in terms of probability distributions on multimolecular structures. Possible generalizations to nonequilibrium statistical mechanics are briefly discussed.     

Grand Molecular Dynamics: A Method for Open Systems

We present a new molecular dynamics method for studying the dynamics of open systems. The method couples a classical system to a chemical potential reservior. In the formulation, following the extended system dynamics approach, we introduce a variable, v to represent the coupling to the chemical potential reservoir. The new variable governs the dynamics of the variation of number of particles in the system. The number of particles is determined by taking the integer part of v. The fractional part of the new variable is used to scale the potential energy and the kinetic energy of an additional particle: i.e., we introduce a fractional particle. We give the ansatz Lagrangians and equations of motion for both the isothermal and the adiabatic forms of grand molecular dynamics. The averages calculated over the trajectories generated by these equations of motion represent the classical grand canonical ensemble (μVT) and the constant chemical potential adiabatic ensemble (μVL) averages, respectively. The microcanonical phase space densities of the adiabatic and isothermal forms the molecular dynamics method are shown to be equivalent to adiabatic constant chemical potential ensemble, and grand canonical ensemble partition functions. We also discuss the extension to multi-component systems, molecular fluids, ionic solutions and the problems and solutions associated with the implementation of the method. The statistical expressions for thermodynamic functions such as specific heat; adiabatic bulk modulus, Grüneissen parameter and number fluctuations are derived. These expressions are used to analyse trajectories of constant chemical potential systems.     

Protein conformational transitions explored by mixed elastic network models

We develop a mixed elastic network model (MENM) to study large-scale conformational transitions of proteins between two (or more) known structures. Elastic network potentials for the beginning and end states of a transition are combined, in effect, by adding their respective partition functions. The resulting effective MENM energy function smoothly interpolates between the original surfaces, and retains the beginning and end structures as local minima. Saddle points, transition paths, potentials of mean force, and partition functions can be found efficiently by largely analytic methods. To characterize the protein motions during a conformational transition, we follow "transition paths" on the MENM surface that connect the beginning and end structures and are invariant to parameterizations of the model and the mathematical form of the mixing scheme. As illustrations of the general formalism, we study large-scale conformation changes of the motor proteins KIF1A kinesin and myosin II. We generate possible transition paths for these two proteins that reveal details of their conformational motions. The MENM formalism is computationally efficient and generally applicable even for large protein systems that undergo highly collective structural changes.