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Register a new userFunctionals linear in curvature and statistics of helical proteins
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The free energy of globular protein chain is considered to be a functional defined on smooth curves in three dimensional Euclidean space. From the requirement of geometrical invariance, together with basic facts on conformation of helical proteins and dynamical characteristics of the protein chains, we are able to determine, in a unique way, the exact form of the free energy functional. Namely, the free energy density should be a linear function of the curvature of curves on which the free energy functional is defined. This model can be used, for example, in Monte Carlo simulations of exhaustive searching the native stable state of the protein chain.
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Nearly a quarter of genomic sequences and almost half of all receptors that are likely to be targets for drug design are integral membrane proteins. Understanding the detailed mechanisms of the folding of membrane proteins is a largely unsolved, key problem in structural biology. Here, we introduce a general model and use computer simulations to study the equilibrium properties and the folding kinetics of a $C_{alpha}$-based two helix bundle fragment (comprised of 66 amino-acids) of Bacteriorhodopsin. Various intermediates are identified and their free energy are calculated toghether with the free energy barrier between them. In 40% of folding trajectories, the folding rate is considerably increased by the presence of non-obligatory intermediates acting as traps. In all cases, a substantial portion of the helices is rapidly formed. This initial stage is followed by a long period of consolidation of the helices accompanied by their correct packing within the membrane. Our results provide the framework for understanding the variety of folding pathways of helical transmembrane proteins.
We study the distribution of first-passage functionals ${cal A}= int_0^{t_f} x^n(t), dt$, where $x(t)$ is a Brownian motion (with or without drift) with diffusion constant $D$, starting at $x_0>0$, and $t_f$ is the first-passage time to the origin. In the driftless case, we compute exactly, for all $n>-2$, the probability density $P_n(A|x_0)=text{Prob}.(mathcal{A}=A)$. This probability density has an essential singular tail as $Ato 0$ and a power-law tail $sim A^{-(n+3)/(n+2)}$ as $Ato infty$. The former is reproduced by the optimal fluctuation method (OFM), which also predicts the optimal paths of the conditioned process for small $A$. For the case with a drift toward the origin, where no exact solution is known for general $n>-1$, the OFM predicts the distribution tails. For $Ato 0$ it predicts the same essential singular tail as in the driftless case. For $Ato infty$ it predicts a stretched exponential tail $-ln P_n(A|x_0)sim A^{1/(n+1)}$ for all $n>0$. In the limit of large Peclet number $text{Pe}= mu x_0/(2D)gg 1$, where $mu$ is the drift velocity, the OFM predicts a large-deviation scaling for all $A$: $-ln P_n(A|x_0)simeqtext{Pe}, Phi_nleft(z= A/bar{A}right)$, where $bar{A}=x_0^{n+1}/{mu(n+1)}$ is the mean value of $mathcal{A}$. We compute the rate function $Phi_n(z)$ analytically for all $n>-1$. For $n>0$ $Phi_n(z)$ is analytic for all $z$, but for $-1<n<0$ it is non-analytic at $z=1$, implying a dynamical phase transition. The order of this transition is $2$ for $-1/2<n<0$, while for $-1<n<-1/2$ the order of transition changes continuously with $n$. Finally, we apply the OFM to the case of $mu<0$ (drift away from the origin). We show that, when the process is conditioned on reaching the origin, the distribution of $mathcal{A}$ coincides with the distribution of $mathcal{A}$ for $mu>0$ with the same $|mu|$.
We present a theoretical model of facilitated diffusion of proteins in the cell nucleus. This model, which takes into account the successive binding/unbinding events of proteins to DNA, relies on a fractal description of the chromatin which has been recently evidenced experimentally. Facilitated diffusion is shown quantitatively to be favorable for a fast localization of a target locus by a transcription factor, and even to enable the minimization of the search time by tuning the affinity of the transcription factor with DNA. This study shows the robustness of the facilitated diffusion mechanism, invoked so far only for linear conformations of DNA.
Protein-fragment seqlets typically feature about 10 amino acid residue positions that are fixed to within conservative substitutions but usually separated by a number of prescribed gaps with arbitrary residue content. By quantifying a general amino acid residue sequence in terms of the associated codon number sequence, we have found a precise modular Fibonacci sequence in a continuous gap-free 10-residue seqlet with either 3 or 4 conservative amino acid substitutions. This modular Fibonacci sequence is genuinely biophysical, for it occurs nine times in the SWISS-Prot/TrEMBL database of natural proteins.
The beautiful structures of single and multi-domain proteins are clearly ordered in some fashion but cannot be readily classified using group theory methods that are successfully used to describe periodic crystals. For this reason, protein structures are considered to be aperiodic, and may have evolved this way for functional purposes, especially in instances that require a combination of softness and rigidity within the same molecule. By analyzing the solved protein structures, we show that orientational symmetry is broken in the aperiodic arrangement of the secondary structural elements (SSEs), which we deduce by calculating the nematic order parameter, $P_{2}$. We find that the folded structures are nematic droplets with a broad distribution of $P_{2}$. We argue that non-zero values of $P_{2}$, leads to an arrangement of the SSEs that can resist mechanical forces, which is a requirement for allosteric proteins. Such proteins, which resist mechanical forces in some regions while being flexible in others, transmit signals from one region of the protein to another (action at a distance) in response to binding of ligands (oxygen, ATP or other small molecules).
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