With the help of force spectroscopy, several analytical theories aim at estimating the rate coefficient of folding for various proteins. Nevertheless, a chief bottleneck lies in the fact that there is still no perfect consensus on how does a force generally perturb the crystal-coil transition. Consequently, the goal of our work is in clarifying the generic behavior of most proteins in force spectroscopy; in other words, what general signature does an arbitrary protein exhibit for its rate coefficient as a function of the applied force? By employing a biomimetic polymer in molecular simulations, we focus on evaluating its respective activation energy for unfolding, while pulling on various pairs of its monomers. Above all, we find that in the vicinity of the force-free scenario, this activation energy possesses a negative slope and a negative curvature as a function of the applied force. Our work is in line with the most recent theories for unfolding, which suggest that such a signature is expected for most proteins, and thus, we further reiterate that many of the classical formulae, that estimate the rate coefficient of the crystal-coil transition, are inadequate. Besides, we also present here an analytical expression which experimentalists can use for approximating the activation energy for unfolding; importantly, it is based on measurements for the mean and variance of the distance between the beads which are being pulled. In summary, our work presents an interesting view for protein folding in force spectroscopy.
Using detailed atomistic simulations,we explore the conformational landscape of aggregates formed by biomimetic antimicrobial(AM) binary methacrylate copolymers,with hydrophobic and charged functional groups and the role of inclusion of polar functional groups on such aggregate morphologies.The effect of sequence of the constituent functional groups on aggregate conformation is also studied by considering random and block sequences along the polymer backbone.Our results suggest that block binary copolymers form large spherical aggregates with effective shielding of hydrophobic groups by charge…
The scales of the white Cyphochilus beetles are endowed with unusual whiteness arising from the exceptional scattering efficiency of their disordered ultrastructure optimized through millions of years of evolution. Here, a simple, one-step method based on water vapor-induced phase separation (VIPS) is developed to prepare ultra-thin polystyrene (PS) films with similar microstructure and comparable optical performance. A typical biomimetic 3.5 um PS film exhibits a diffuse reflectance of 61% at 500 nm, which translates into a transport mean free path below 1 um. A complete optical characterizat…
Kinetics of conformational change of a semiflexible polymer under mechanical external field were investigated with Langevin dynamics simulations. It is found that a semiflexible polymer exhibits large hysteresis in mechanical folding/unfolding cycle even with a slow operation, whereas in a flexible polymer, the hysteresis almost disappears at a sufficiently slow operation. This suggests that the essential features of the structural transition of a semiflexible polymer should be interpreted at least on a two-dimensional phase space. The appearance of such large hysteresis is discussed in relati…
The dynamics of polymer decompression, i.e., a process from compressed, compact state to the relaxed swoll en conformation, can be formally described as a {\it nonlinear diffusion}. We discuss here two basic examples: (i) the expansion, or unfolding from a compact state, and (ii) the ejection of pressed polymer through a pore. The problem can be solved exactly for the case (i), but not for the latter. Even in such situations, a scheme called {\it uniform approximation} is shown to be useful to get a physical insight involved. Its application to the case (ii) is able to account for conflicting …
We theoretically investigate the kinetics of the folding transition of a single semiflexible polymer. In the folding transition, the growth rate decrease with an increase in the number of monomers in a collapsed domain, suggesting that the main contribution to dissipation is from the motion of the domain. In the unfolding transition, dynamic scaling exponents, 1/8 and 1/4, were determined for disentanglement and relaxation steps, respectively. We performed Langevin dynamics simulations to test our theory. It is found that our theory is in good agreement with simulations. We also propose the ki…
Effect of molecular crowding and confinement experienced by protein in the cell during unfolding has been studied by modeling a linear polymer chain on a percolation cluster. It is known that internal structure of the cell changes in time, however, they do not change significantly from their initial structure. In order to model this we introduce the correlation among the different disorder realizations. It was shown that the force-extension behavior for correlated disorder in both constant force ensemble (CFE) and constant distance ensemble (CDE) is significantly different than the one obtaine…
A statistical mechanical description of flexible and semi-flexible polymer chains in a poor solvent is developed in the constant force and constant distance ensembles. We predict the existence of many intermediate states at low temperatures stabilized by the force. A unified response to pulling and compressing forces has been obtained in the constant distance ensemble. We show the signature of a cross-over length which increases linearly with the chain length. Below this cross-over length, the critical force of unfolding decreases with temperature, while above, it increases with temperature. F…
The recently proposed fluctuation relation in unfolding forces [Phys. Rev. E ${\bf 84}$, $060101$(R) $(2011)$] is re-examined taking into account the explicit time dependence of the force distribution. The stretching of a tethered Rouse polymer is exactly solved and the ratio of the probabilities of positive to negative forces is shown to be an exponential in force. Extensive steered molecular dynamics simulations of unfolding of deca alanine peptide confirm the form of fluctuation relation proposed earlier, but with explicit correct time dependence of unfolding forces taken into account. From…
We study the conformations of polymer chains in a poor solvent, with and without bending rigidity, by means of a simple statistical mechanics model. This model can be exactly solved for chains of length up to N=55 using exact enumeration techniques. We analyze in details the differences between the constant force and constant distance ensembles for large but finite N. At low temperatures, and in the constant force ensemble, the force-extension curve shows multiple plateaus (intermediate states), in contrast with the abrupt transition to an extended state prevailing in the $N \to \infty$ limit.…
We consider a stiff polymer chain in poor solvent and apply a force at one end of the chain. We find that by varying the stiffness parameter, polymer undergoes a transition from the globule state to the folded like state. The conformation of folded state mimics the $β$-sheet as seen in titin molecule. Using exact enumeration technique, we study the extension-force and force-temperature diagrams of such a system. Force-temperature diagram shows the re-entrance behaviour for flexible chain. However, for stiff chain this re-entrance behaviour is absent and there is an enhancement in $θ$-temperatu…
The 3D organisation of the genome in interphase cells is not a randomly folded polymer. Rather, experiments show that chromosomes arrange into a network of 3D compartments that correlate with biological processes, such as transcription, chromatin modifications, and protein binding. However, these compartments do not exist during cell division when the DNA is condensed, and it is unclear how and when they emerge. In this paper, we focus on the early stages after cell-division as the chromosomes start to decondense. We use a simple polymer model to understand the types of 3D structures that emer…
We summarize the recent simulation progress of micromanipulation experiments on RNAs. Our work mainly consults with two important small RNAs unfolding experiments carried out by Bustamante group. Our results show that, in contrast to protein cases, using the single polymer elastic theory and the well known RNA secondary structure free energy knowledge, we can successively simulate various behaviors of force unfolding RNAs under different experimental setups from equilibrium to far-from equilibrium. Particularly, our simulation would be helpful in understanding Jarzynski's remarkable equality, …
We describe a model of cytoskeletal mechanics based on the force-induced conformational change of protein cross-links in a stressed polymer network. Slow deformation of simulated networks containing cross-links that undergo repeated, serial domain unfolding leads to an unusual state--with many cross-links accumulating near the critical force for further unfolding. Thermal activation of these links gives rise to power-law rheology resembling the previously unexplained mechanical response of living cells. Moreover, we hypothesize that such protein cross-links function as biochemical mechano-sens…
We study the force-induced unfolding of a homopolymer on the three dimensional Sierpinski gasket. The polymer is subject to a contact energy between nearest neighbour sites not consecutive along the chain and to a stretching force. The hierarchical nature of the lattice we consider allows for an exact treatment which yields the phase diagram and the critical behaviour. We show that for this model mean field predictions are not correct, in particular in the exact phase diagram there is {\em not} a low temperature reentrance and we find that the force induced unfolding transition below the theta…
We propose the quantitative mean-field theory of mechanical unfolding of a globule formed by long flexible homopolymer chain collapsed in poor solvent and subjected to an extensional force We show that with an increase in the applied force the globule unfolds as a whole without formation of an intermediate state. The value of the threshold force and the corresponding jump in the distance between chain ends increase with a deterioration of the solvent quality and / or with an increase in the degree of polymerization. This way of globule unfolding is compared with that in the D-ensemble, when th…
We report on the nonlinear mechanical properties of a statistically homogeneous, isotropic semiflexible network cross-linked by polymers containing numerous small unfolding domains, such as the ubiquitous F-actin cross-linker Filamin. We show that the inclusion of such proteins has a dramatic effect on the large strain behavior of the network. Beyond a strain threshold, which depends on network density, the unfolding of protein domains leads to bulk shear softening. Past this critical strain, the network spontaneously organizes itself so that an appreciable fraction of the Filamin cross-linker…
Using self-organized polymer models, we predict mechanical unfolding and refolding pathways of ribo-zymes, and the green fluorescent protein. In agreement with experiments, there are between six and eight unfolding transitions in the Tetrahymena ribozyme. Depending on the loading rate, the number of rips in the force-ramp unfolding of the Azoarcus ribozymes is between two and four. Force-quench refolding of the P4-P6 subdomain of the Tetrahymena ribozyme occurs through a compact intermediate. Subsequent formation of tertiary contacts between helices P5b-P6a and P5a/P5c-P4 leads to the native s…
Mechanical unfolding of RNA structures, ranging from hairpins to ribozymes, using laser optical tweezer (LOT) experiments have begun to reveal the features of the energy landscape that cannot be easily explored using conventional experiments. Upon application of constant force ($f$), RNA hairpins undergo cooperative transitions from folded to unfolded states whereas subdomains of ribozymes unravel one at a time. Here, we use a self-organized polymer (SOP) model and Brownian dynamics simulations to probe mechanical unfolding at constant force and constant-loading rate of four RNA structures of …
We study the structural and dynamical behaviors of a diblock copolymer chain in a bath of active Brownian particles (ABPs) by extensive Brownian dynamics simulation in a two-dimensional model system. Specifically, the A block of chain is self-attractive, while the B block is self-repulsive. We find, beyond a threshold, the A block unfolds with a pattern like extracting a woolen string from a ball. The critical force decreases with the increase of the B block length,NB, for short cases, then keeps a constant with further increase of NB. In addition, we find a power law exists between the unfold…
Liquid-liquid phase separation (LLPS) is an emerging and universal mechanism for intracellular biomolecule organization, particularly, via the formation of membraneless organelles (MOs). Intrinsically disordered proteins (IDPs) are the main constituents of MOs, wherein multivalent interactions and low-complexity domains (LCDs) drive LLPS. Using short oligopeptide derived from LCDs as 'stickers' and dextran backbones as 'spacers', we designed polymer-oligopeptide hybrids to mimic the multivalent FUS protein as represented by the 'stickers-and-spacers' model. We demonstrated that hybrids underwe…
The mechanical stretching of single poly-proteins is an emerging tool for the study of protein (un)folding, chemical catalysis and polymer physics at the single molecule level. The observed processes i.e unfolding or reduction events, are typically considered to be stochastic and by its nature are susceptible to be censored by the finite duration of the experiment. Here we develop a formal analytical and experimental description on the number of observed events under various conditions of practical interest. We provide a rule of thumb to define the experiment protocol duration. Finally we prov…
The stability of a $β$-sheeted conformation and its transition into a random coil are studied with a 2D lattice biopolymer model. At low temperature and low external force, the polymer folds back and forth on itself and forms a $β$-sheet. Our analytical calculation and Monte Carlo simulation reveal that a co-operative $β$-sheet--random coil transition takes places when the temperature or force is increased, with a dramatic decrease in the contact number. These predictions are in good agreement with experiments on titin protein. This transition is not a real phase-transition, indicating that ba…
The unfolding of a polymer below the $θ$ point when pulled by an external force is studied both in d=2 on the lattice and in $d=3$ off lattice. A ground state analysis of finite length chains shows that the globule unfolds via multiple steps, corresponding to transitions between different minima, in both cases. In the infinite length limit, these intermediate minima have a qualitative effect only in $d=2$. The phase diagram in d=2 is determined using transfer matrix techniques. Energy-entropy and renormalization group arguments are given which predict a qualitatively correct phase diagram and …
The experimental calorimetric two-state criterion requires the van't Hoff enthalpy $ΔH_{\rm vH}$ around the folding/unfolding transition midpoint to be equal or very close to the calorimetric enthalpy $ΔH_{\rm cal}$ of the entire transition. We use an analytical model with experimental parameters from chymotrypsin inhibitor 2 to elucidate the relationship among several different van't Hoff enthalpies used in calorimetric analyses. Under reasonable assumptions, the implications of these $ΔH_{\rm vH}$'s being approximately equal to $ΔH_{\rm cal}$ are equivalent: Enthalpic variations among denatu…
Using coarse-grained model we have explored forced-unfolding of RNA hairpin as a function of $f_S$ and the loading rate ($r_f$). The simulations and theoretical analysis have been done without and with the handles that are explicitly modeled by semiflexible polymer chains. The mechanisms and time scales for denaturation by temperature jump and mechanical unfolding are vastly different. The directed perturbation of the native state by $f_S$ results in a sequential unfolding of the hairpin starting from their ends whereas thermal denaturation occurs stochastically. From the dependence of the unf…
Equilibrium mechanical unfolding of a globule formed by long flexible homopolymer chain collapsed in a poor solvent and subjected to an extensional force f (force-clamp mode) or extensional deformation D (position-clamp mode) is studied theoretically. Our analysis, like all previous analysis of this problem, shows that the globule behaves essentially differently in two modes of extension. In the force-clamp mode, mechanical unfolding of the globule with increasing applied force occurs without intramolecular microphase segregation, and at certain threshold value of the pulling force the globule…
Single molecule force spectroscopy provide details of the underlying energy surfaces of proteins which are essential to the understanding of their unfolding process. Recently, it has been observed experimentally that by pulling proteins in different directions relative to their secondary structure, one can gain a better understanding of the shape of the energy landscape. We consider simple lattice models which are anisotropic in nature to study the response of a force in unfolding of a polymer. Our analytical solution of the model, supported by extensive numerical calculations, reveal that the…
The interior of cells is crowded thus making it important to assess the effects of macromolecules on the folding of proteins. Using the Self-Organized Polymer (SOP) model, which is a coarse-grained representation of polypeptide chains, we probe the mechanical stability of Ubiquitin (Ub) monomers and trimers ((Ub)$_3$) in the presence of monodisperse spherical crowding agents. Crowding increases the volume fraction ($Φ_c$)-dependent average force ($<f_u(Φ_c)>$), relative to the value at $Φ_c = 0$, needed to unfold Ub and the polyprotein. For a given $Φ_c$, the values of $<f_u(Φ_c)>$ increase as…
Hypotheses: Additives are commonly used to tune macromolecular conformational transitions. Among additives, trehalose is an excellent bioprotectant and among responsive polymers, PNIPAM is the most studied material. Nevertheless, their interaction mechanism so far has only been hinted without direct investigation, and, crucially, never elucidated in comparison to proteins. Detailed insights would help understand to what extent PNIPAM microgels can effectively be used as synthetic biomimetic materials, to reproduce and study, at the colloidal scale, isolated protein behavior and its sensitivity…
In order to understand the nuclei which develop during the course of protein folding and unfolding, we examine phase segregation of a single heteropolymer chain which occurs in equilibrium. These segregated conformations are characterized by a nucleus of monomers which are superimposable upon the native conformation. We computationally generate the phase segregation by applying a ``folding pressure,'' or adding an energetic bonus for native monomer-monomer contacts. The computer models reveal a fundamental difference in the nucleation process between heteropolymeric and the more familiar vapor…
Folding and unfolding of biopolymers are often manipulated in experiment by tuning pH, temperature, single-molecule force or shear field. Here we carry out Brownian dynamics simulations to explore the behavior of a single self-attracting chain in the suspension of self-propelling particles (SPPs). As the propelling force increases, globule-stretch (G-S) transition of the chain happens due to the enhanced disturbance from SPPs. Two distinct mechanisms of the transition in the limits of low and high rotational diffusion rates of SPPs have been observed: shear effect at low rate and collision-ind…
The HP model of protein folding, where the chain exists in a free medium, is investigated using a parallel Monte Carlo scheme based upon Wang-Landau sampling. Expanding on the work of Wust and Landau by introducing a lesser known replica -exchange scheme between individual Wang- Landau samplers, the problem of dynamical trapping (spiking in the density of states) was avoided and an enhancement in the efficiency of traversing configuration space was obtained. Highlighting dynamical trapping as an issue for lattice polymer simulations for increasing lengths is explicitly done here for the first …
In contrast to proteins much less attention has been focused on development of computational models for describing RNA molecules, which are being recognized as playing key roles in many cellular functions. Current atomically detailed force fields are not accurate enough to capture the properties of even simple nucleic acid constructs. In this article, we review our efforts to develop coarse-grained (CG) models that capture the underlying physics for the particular length scale of interest. Two models are discussed. One of them is the Three Interaction Site (TIS) model in which each nucleotide …
We present a panoramic view of the utility of coarse-grained (CG) models to study folding and functions of proteins and RNA. Drawing largely on the methods developed in our group over the last twenty years, we describe a number of key applications ranging from folding of proteins with disulfide bonds to functions of molecular machines. After presenting the theoretical basis that justifies the use of CG models, we explore the biophysical basis for the emergence of a finite number of folds from lattice models. The lattice model simulations of approach to the folded state show that non-native int…
