Curved polymer and polyelectrolyte brushes beyond the Daoud-Cotton model

The resolution of Levinthal’s paradox concerning the ability of proteins to fold rapidly postulates the existence of a rough “folding funnel” in free energy space that guides the protein to its lowest free energy, native state. To study the folding of the protein ribonuclease A we have mapped it onto a 124 monomer, coarse-grained HP lattice model and onto an H0P model that also includes “neutral” 0-mers in addition to the hydrophobic H-mers and polar P-mers. Using Replica Exchange Wang-Landau sampling, we determined the density of states, g(E), which we then used to calculate the free energy of the protein vs end-to-end distance as a function of temperature. At low temperature the HP model shows a rather shallow and at free energy minimum, while the H0P model maintains a rough free energy funnel. Unlike the common, schematic figures, we find an asymmetric folding funnel that also changes shape substantially as the temperature decreases. Even the location of the free energy minimum shifts as the temperature decreases.

The reversible binding between a planar polymer layer functionalized by ligands and a planar cell surface containing different densities of mobile receptors has been studied by Monte Carlo simulations. Using the acceptance-ratio method, the distance-dependent profiles for the average number of ligands bound to receptors, the total free energy for the polymer layer-cell surface interaction and the interaction force were obtained. Four main design parameters for the polymer layer were considered: the degree of functionalization, chain degree of polymerization, polymer grafting density and the binding energy for the ligand-receptor interaction. We found that an increase in the degree of functionalization or in the absolute energy of ligand-receptor binding results in a larger number of ligands bound to the receptors, lower free energy, and stronger attractive force. Polymer layers composed of shorter chains were found to exhibit a deeper and narrower free energy profile and a larger attractive force, while longer tethers can interact with the cell surface at a larger and broader range of separation distances, in agreement with experimental observations. Our simulation results show that the increase in polymer grafting density from the mushroom to brush regime enhances the ligand availability and results in a stronger attractive force, increases the maximum binding distance, but exhibits a shallower free energy minimum due to the smaller tolerance to compression for polymer layers with high grafting density. We used two measures of the polymer layer binding affinity to the cell surface: the free energy minimum, related to the equilibrium binding constant and the fraction of bound ligands. We found that the polymer layers with a smaller chain length and grafting density, larger degree of functionalization, and larger absolute binding energy exhibit both a larger equilibrium binding constant to the cell surface and a larger average number of bound ligands, except for high binding energies when the maximum level of binding is reached independently of polymer length and grafting density. We showed that high binding specificity can be achieved by the polymer layers with intermediate ligand-receptor binding energies or an intermediate number of ligands, as a larger binding energy or number of ligands ensures a high binding affinity but lacks specificity while a smaller binding energy or number of ligands provides inadequate affinity. We found that the results for polymer layers with different properties follow a similar pattern when both high binding affinity to cells with high receptor density and high binding specificity are considered. As a result, the optimal design of the polymer layers can be achieved by using several different strategies, which are discussed.

We present an analytical self-consistent-field (SCF) theory for a neutral polymer brush (a layer of long polymer chains end-grafted to a surface) with annealed excluded volume interactions between the monomer units. This model mimics the reversible adsorption of solute molecules or aggregates, such as small globular proteins or surfactant micelles, on the grafted chains. The equilibrium structural properties of the brush (the brush thickness, the monomer density profile, the distribution of the end segments of the grafted chains) as well as the overall adsorbed amount and the adsorbate density profile are analyzed as a function of the grafting density, the excluded volume parameters and the chemical potential (the concentration) of the adsorbate in the solution. We demonstrate that, when the grafting density is varied, the overall adsorbed amount always exhibits a maximum, whereas the root-mean-square brush thickness either increases monotonically or passes through a (local) minimum. At high grafting densities the chains are loaded by adsorbed aggregates preferentially in the distal region of the brush, whereas in the region proximal to the grafting surface depletion of aggregates occurs and the polymer brush retains an unperturbed structure. Depending on the relative strength of the excluded volume interactions between unloaded and loaded monomers both the degree of loading of the chains and the polymer density profile are either continuous or they exhibit a discontinuity as a function of the distance from the grafting surface. In the latter case intrinsic phase separation occurs in the brush: the dense phase consists of unloaded and weakly extended chains and occupies the region proximal to the surface, whereas a more dilute phase consisting of highly loaded and strongly extended chains forms the periphery of the brush.

ABSTRACTThe shape of a liquid drop on a flat solid surface is examined by using the minimization of free energy. This free energy consists of the contributions from the gravitational potential energy of the drop mass and the energy of the interfacial surfaces. The minimization of free energy is solved by the Lagrange multipliers method keeping the drop volume constant. The generic differential equation for the shape of the drop and a relation among the contact angles (Young equation) are determined as a result of the minimization via the variation of a functional and by imposing the condition for equilibrium of the drop. The differential equation is solved analytically around the origin of the selected coordinate axis and for two special cases. For the general solution, the differential equation is solved numerically to obtain the general shape of the sessile drop. Furthermore, a new Axisymmetric Drop Shape Analysis method is proposed for non–wetting sessile drop.

BackgroundRNA inverse folding is the problem of finding one or more sequences that fold into a user-specified target structure s0, i.e. whose minimum free energy secondary structure is identical to the target s0. Here we consider the ensemble of all RNA sequences that have low free energy with respect to a given target s0.ResultsWe introduce the program RNAdualPF, which computes the dual partition functionZ∗, defined as the sum of Boltzmann factors exp(−E(a,s0)/RT) of all RNA nucleotide sequences a compatible with target structure s0. Using RNAdualPF, we efficiently sample RNA sequences that approximately fold into s0, where additionally the user can specify IUPAC sequence constraints at certain positions, and whether to include dangles (energy terms for stacked, single-stranded nucleotides). Moreover, since we also compute the dual partition functionZ∗(k) over all sequences having GC-content k, the user can require that all sampled sequences have a precise, specified GC-content.Using Z∗, we compute the dual expected energy 〈E∗〉, and use it to show that natural RNAs from the Rfam 12.0 database have higher minimum free energy than expected, thus suggesting that functional RNAs are under evolutionary pressure to be only marginally thermodynamically stable.We show that C. elegans precursor microRNA (pre-miRNA) is significantly non-robust with respect to mutations, by comparing the robustness of each wild type pre-miRNA sequence with 2000 [resp. 500] sequences of the same GC-content generated by RNAdualPF, which approximately [resp. exactly] fold into the wild type target structure. We confirm and strengthen earlier findings that precursor microRNAs and bacterial small noncoding RNAs display plasticity, a measure of structural diversity.ConclusionWe describe RNAdualPF, which rapidly computes the dual partition functionZ∗ and samples sequences having low energy with respect to a target structure, allowing sequence constraints and specified GC-content. Using different inverse folding software, another group had earlier shown that pre-miRNA is mutationally robust, even controlling for compositional bias. Our opposite conclusion suggests a cautionary note that computationally based insights into molecular evolution may heavily depend on the software used.C/C++-software for RNAdualPF is available at http://bioinformatics.bc.edu/clotelab/RNAdualPF.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-016-1280-6) contains supplementary material, which is available to authorized users.

A computer method for construction of secondary structure from polynucleotide sequence. Possible structure of the bacterial replication origin

Accurately computing the free energy for biological processes like protein folding or protein-ligand association remains a challenging problem. Both describing the complex intermolecular forces involved and sampling the requisite configuration space make understanding these processes innately difficult. Herein, we address the sampling problem using a novel methodology we term "movable type". Conceptually it can be understood by analogy with the evolution of printing and, hence, the name movable type. For example, a common approach to the study of protein-ligand complexation involves taking a database of intact drug-like molecules and exhaustively docking them into a binding pocket. This is reminiscent of early woodblock printing where each page had to be laboriously created prior to printing a book. However, printing evolved to an approach where a database of symbols (letters, numerals, etc.) was created and then assembled using a movable type system, which allowed for the creation of all possible combinations of symbols on a given page, thereby, revolutionizing the dissemination of knowledge. Our movable type (MT) method involves the identification of all atom pairs seen in protein-ligand complexes and then creating two databases: one with their associated pairwise distant dependent energies and another associated with the probability of how these pairs can combine in terms of bonds, angles, dihedrals and non-bonded interactions. Combining these two databases coupled with the principles of statistical mechanics allows us to accurately estimate binding free energies as well as the pose of a ligand in a receptor. This method, by its mathematical construction, samples all of configuration space of a selected region (the protein active site here) in one shot without resorting to brute force sampling schemes involving Monte Carlo, genetic algorithms or molecular dynamics simulations making the methodology extremely efficient. Importantly, this method explores the free energy surface eliminating the need to estimate the enthalpy and entropy components individually. Finally, low free energy structures can be obtained via a free energy minimization procedure yielding all low free energy poses on a given free energy surface. Besides revolutionizing the protein-ligand docking and scoring problem this approach can be utilized in a wide range of applications in computational biology which involve the computation of free energies for systems with extensive phase spaces including protein folding, protein-protein docking and protein design.

We develop a theory for the equilibrium concentration profile formed by a compressible polymer melt near a solid wall (or the mathematically equivalent incompressible polymer solution near a solid wall). The theory uses a Landau–Ginzburg free energy functional with a concentration dependent square gradient coefficient and an additional local contribution characterizing the influence of the wall. We introduce a mean-field algorithm for constructing the surface free energy contribution from the expression for the bulk free energy of lattice polymers. This algorithm automatically includes both energetic and entropic contributions with no adjustable parameters for lattice systems and can be applied for branched polymers as well. Approximate analytical solutions are provided for one-phase polymer density profiles at neutral, repulsive, and attractive walls. The approximate solutions reflect the behavior of the numerical solutions and display (only at lower polymer densities) a near wall linear variation of the polymer density which crosses over to an exponential approach to the bulk concentration. The numerically evaluated profiles for both athermal and nonathermal melts compare well with available Monte Carlo simulation data for a neutral wall. Physical arguments are presented which anticipate the existence of deviations between theory and simulations at higher densities. The use of lattice cluster theory free energy functions enables us for the first time to investigate the dependence of the density profile on the polymer architecture.

The minimum free energy (MFE) of ribonucleic acids (RNAs) increases at an apparent linear rate with sequence length. Simple indices, obtained by dividing the MFE by the number of nucleotides, have been used for a direct comparison of the folding stability of RNAs of various sizes. Although this normalization procedure has been used in several studies, the relationship between normalized MFE and length has not yet been investigated in detail. Here, we demonstrate that the variation of MFE with sequence length is not linear and is significantly biased by the mathematical formula used for the normalization procedure. For this reason, the normalized MFEs strongly decrease as hyperbolic functions of length and produce unreliable results when applied for the comparison of sequences with different sizes. We also propose a simple modification of the normalization formula that corrects the bias enabling the use of the normalized MFE for RNAs longer than 40 nt. Using the new corrected normalized index, we analyzed the folding free energies of different human RNA families showing that most of them present an average MFE density more negative than expected for a typical genomic sequence. Furthermore, we found that a well-defined and restricted range of MFE density characterizes each RNA family, suggesting the use of our corrected normalized index to improve RNA prediction algorithms. Finally, in coding and functional human RNAs the MFE density appears scarcely correlated with sequence length, consistent with a negligible role of thermodynamic stability demands in determining RNA size.

5 - Thermodynamics of solids

Self-assembly of (A2B2)5 multigraft block copolymer: The length scale and phase transition

Minimization of free energy is used to calculate the equilibrium shapes of sessile liquid drops under different wetting conditions and under the influence of a gravitational body force. The total system free energy is assumed to obey an equation recently derived by Searcy [1], and the shapes of sessile drops of different sizes are calculated and compared with experimental measurements. Stationary states in the free energy are found using standard variational principles, which lead not only to an Euler-Lagrange differential equation for the drop shape, but also to the boundary conditions at the three-phase junction where the liquid, solid, and surrounding vapor meet. The analysis shows that the Searcy equation leads to the classical Young-Dupr equation for the thermodynamic contact angle. Drop shapes calculated by this approach are in excellent agreement with experimental observations of drops of pure water resting on a clean polytetrafluoroethylene (PTFE) substrate.

Dynalign: an algorithm for finding the secondary structure common to two RNA sequences

The asymmetry between the left and right sides seems to be a general principle of organization of the nervous systems in Bilateria, providing the foundations for a plethora of leftward and rightward biases in behaviour as documented in species ranging from Caenorhabditis elegans nematodes to humans. Several theories have been put forward to account for the existence and maintenance in the evolution of the asymmetric organization of the brain at both individual and population levels. However, what is missing in theorizing about the evolution of brain asymmetry is an overarching general hypothesis that may subsume all different aspects of current models. Here, we tried to provide an overarching general framework based on the energy and free-energy minimization principle, which proved so valuable in other areas of neuroscience. We found that at the individual level the antisymmetric singlet configuration realizes the lowest energy state of the system, whereas at the group level, the spontaneous emergence of directional asymmetry arises as a consequence of the minimization of the free energy of the system, which guarantees its stability and equilibrium. We thus argue that the various phenomenological aspects of brain asymmetry that have been captured in biology-e.g. sparing of neural tissue, control of unitary motor responses and, at the population level, evolutionarily stable strategies described by mathematical games theory-may be thought of as the manifestation of a more general principle of energy minimization generating, among others, asymmetry of the brains.

In an attempt to improve automated gene prediction in the untranslated region of a gene, we completed an in-depth analysis of the minimum free energy for 8,689 sub-genetic DNA sequences. We expanded Zhang's classification model and classified each sub-genetic sequence into one of 27 possible motifs. We calculated the minimum free energy for each motif to explore statistical features that correlate to biologically relevant sub-genetic sequences. If biologically relevant sub-genetic sequences fall into distinct free energy quanta it may be possible to characterize a motif based on its minimum free energy. Proper characterization of motifs can lead to greater understanding in automated genefinding, gene variability and the role DNA structure plays in gene network regulation. Our analysis determined: (1) the average free energy value for exons, introns and other biologically relevant sub-genetic sequences, (2) that these subsequences do not exist in distinct energy quanta, (3) that introns exist however in a tightly coupled average minimum free energy quantum compared to all other biologically relevant sub-genetic sequence types, (4) that single exon genes demonstrate a higher stability than exons which span the entire coding sequence as part of a multi-exon gene and (5) that all motif types contain a free energy global minimum at approximately nucleotide position 1,000 before reaching a plateau. These results should be relevant to the biochemist and bioinformatician seeking to understand the relationship between sub-genetic sequences and the information behind them.