Investigation of the folding profiles of evolutionarily selected model proteins

Statistical thermodynamics of liquid-liquid phase separation in ternary systems during complex coacervation

Liquid-liquid phase separation leading to complex coacervation in a ternary system (oppositely charged polyion and macroion in a solvent) is discussed within the framework of a statistical thermodynamics model. The polyion and the macroion in the ternary system interact to form soluble aggregates (complexes) in the solvent, which undergoes liquid-liquid phase separation. Four necessary conditions are shown to drive the phase separation: (i) (σ{23}){3}r/Φ{23c}≥(64/9α{2})(χ{23}Φ{3}){2} , (ii) r≥[64(χ{23}Φ{3}){2}/9α{2}σ{23}{3}]{1/2}, (iii) χ{23}≥(2χ{231}-1)/Φ{23c}Φ{3}, and (iv) (σ{23}){2}/sqrt[I]≥8/3α(2χ{231}-1) (where σ{23} is the surface charge on the complex formed due to binding of the polyelectrolyte and macroion, Φ{23c} is the critical volume fraction of the complex, χ{23} is the Flory interaction parameter between polyelectrolyte and macroion, χ{231} is the same between solvent and the complex, Φ{3} is the volume fraction of the macroions, I is the ionic strength of the solution, α is electrostatic interaction parameter and r is typically of the order of molecular weight of the polyions). It has been shown that coacervation always requires a hydrated medium. In the case of a colloidal macroion and polyelectrolyte coacervation, molecular weight of polyelectrolyte must satisfy the condition r≥10{3} Da to exhibit liquid-liquid phase separation. This model has been successfully applied to study the coacervation phenomenon observed in aqueous Laponite (macroion)-gelatin (polyion) system where it was found that the coacervate volume fraction, δΦ{23}∼χ{231}{2} (where δΦ{23} is the volume fraction of coacervates formed during phase separation). The free energy and entropy of this process have been evaluated, and a free-energy landscape has been drawn for this system that maps the pathway leading to phase separation.

Neutrality and evolvability of designed protein sequences

The effect of foldability on protein's evolvability is analyzed by a two-prong approach consisting of a self-consistent mean-field theory and Monte Carlo simulations. Theory and simulation models representing protein sequences with binary patterning of amino acid residues compatible with a particular foldability criteria are used. This generalized foldability criterion is derived using the high temperature cumulant expansion approximating the free energy of folding. The effect of cumulative point mutations on these designed proteins is studied under neutral condition. The robustness, protein's ability to tolerate random point mutations is determined with a selective pressure of stability (ΔΔG) for the theory designed sequences, which are found to be more robust than that of Monte Carlo and mean-field-biased Monte Carlo generated sequences. The results show that this foldability criterion selects viable protein sequences more effectively compared to the Monte Carlo method, which has a marked effect on how the selective pressure shapes the evolutionary sequence space. These observations may impact de novo sequence design and its applications in protein engineering.

Free-energy landscape of alcohol driven coacervation transition in aqueous gelatin solutions

Liquid-liquid phase separation of a homogeneous polyampholyte (gelatin) solution into a dense polymer-rich coacervate and the dilute supernatant phase is discussed through free-energy landscape formalism. We have evaluated the free energy and entropy of the system as it undergoes the phenomenon of simple coacervation, driven by the addition of a nonsolvent. Electrophoretic mobility (mu) and turbidity measurements were performed on 0.01% and 0.05% (w/v) aqueous gelatin solutions that were driven towards coacervation by the addition of ethanol. The mobility of the polyampholyte molecules, which was typically mu approximately 0.38+/-0.02 microm/s cm/V in water, gradually reduced for the soluble intermolecular complexes to a plateau value of mu approximately 0.11+/-0.01 microm/s cm/V as the ethanol volume fraction equaled phi(ns) approximately 0.47+/-0.03, which coincided with the first appearance of coacervate droplets (coacervation transition) observed from turbidity measurements, a behavior found to be invariant of gelatin concentration. These results were used as input to the theoretical model to explicitly construct the free-energy landscape for a single gelatin chain and the global system comprising the polymer-rich coacervate and the dilute supernatant phase.

Local-structural diversity and protein folding: Application to all-β off-lattice protein models

Global measures of structural diversity within a distribution of biopolymers, such as the radius of gyration and percent native contacts, have proven useful in the analysis of simulation data for protein folding. In this paper we describe a statistical-based methodology to quantify the local structural variability of a distribution of biopolymers, applied to 46- and 69-"residue" off-lattice, three-color model proteins. Each folds into beta-barrel structures. First we perform a principal component analysis of all interbead distance variables for a large number of independent, converged Boltzmann-distributed samples of conformations collected at each of a wide range of temperatures. Next, the principal component vectors are subjected to orthogonal (varimax) rotation. The results are displayed on so-called "squared-loading" plots. These provide a quantitative measure of the contribution to the sample variance of the position of each residue relative to the others. Dominant structural elements, those having the largest structural diversity within the sampled distribution, are responsible for peaks and shoulders observed in the specific heat versus temperature curves, generated using the weighted histogram analysis method. The loading plots indicate that the local-structural diversity of these systems changes gradually with temperature through the folding transition but radically changes near the collapse transition temperature. The analysis of the structural overlap order statistic suggests that the 46-mer thermodynamic folding transition involves the native state and at least three other nearly native intermediates. In the case of the 46-mer protein model, data are generated at sufficiently low temperatures that squared-loading plots, coupled with cluster analysis, provide a local and energetic description of its glassy state.