Partition Function Zeros and Finite Size Scaling of Helix-Coil Transitions in a Polypeptide

Abstract

We report on multicanonical simulations of the helix-coil transition of a polypeptide. The nature of this transition was studied by calculating partition function zeros and the finite-size scaling of various quantities. New estimates for critical exponents are presented.87.15. He, 64.70Cn, 02.50.Ng A common, ordered structure in proteins is the α-helix and it is conjectured that formation of α-helices is a key factor in the early stages of protein-folding [1]. It is long known that α-helices undergo a sharp transition towards a random coil state when the temperature is increased. The characteristics of this so-called helix-coil transition have been studied extensively [2], most recently in Refs. [3,4]. They are usually described in the framework of Zimm-Bragg-type theories [5] in which the homopolymers are approximated by a one-dimensional Ising model with the residues as "spins" taking values "helix" or "coil", and solely local interactions. Hence, in such theories thermodynamic phase transitions are not possible. However, in preliminary work [4] it was shown that our all-atom model of poly-alanine exhibits a phase transition between the ordered helical state and the disordered random-coil state. It was conjectured that this transition is due to long range interactions in our model and the fact that it is not one-dimensional: it is known that the 1D Ising model with long-range interactions also exhibits a phase transition at finite T if the interactions decay like 1/r σ with 1 ≤ σ < 2 [6]. Our aim now is to investigate this transition in the frame work of a critical theory by means of the finite size scaling (FSS) analysis of partition function zeros. Analysis of partition function zeros is a well-known tool in the study of phase transitions, but was to our knowledge never used before to study biopolymers.For our project, the use of the multicanonical algorithm [7] was crucial. The various competing interactions within the polymer lead to an energy landscape characterized by a multitude of local minima. Hence, in the low-temperature region, canonical simulations will tend to get trapped in one of these minima and the simulation will not thermalize within the available CPU time. One standard way to overcome this problem is the application of the multicanonical algorithm [7] and other generalized-ensemble techniques [8] to the protein folding problem [9]. For poly-alanine, both the failure of standard Monte Carlo techniques and the superior performance of the multicanonical algorithm are extensively documented in earlier work [10].In the multicanonical algorithm [7] conformations with energy E are assigned a weight w mu (E) ∝ 1/n(E). Here, n(E) is the density of states. A simulation with this weight will lead to a uniform distribution of energy:This is because the simulation generates a 1D random walk in the energy, allowing itself to escape from any local minimum. Since a large range of energies are sampled, one can use the reweighting techniques [11] to calculate thermodynamic quantities over a wide range of...

Helix formation and folding in an artificial peptide

“…In previous work [23,8,9] we could show that polyalanine has a pronounced transition between a disordered coil phase and an ordered state in which the polymer forms an α-helix. For this reason, we expect formation of α-helices in our peptide, and the average number of helical residues < n H > is therefore one of the quantities that we have measured.…”

“…We have neglected in the simulations the interaction of our artifical peptide with the surrounding solvent. While this is certainly a crude approximation, it allows us not only to relate our results to our previous studies on helixcoil transition in poly-alanine that also relied on gas-phase simulations [8][9][10], but also to study the extend to that secondary structure formation and folding are determined by intrinsic properties of the peptide. Our data suggest that the peptide in gas-phase folds in a two-step process: in a first step two α-helices are formed in what amounts to a first order transition.…”

“…The characteristics of this so-called helix-coil transition have been studied extensively [6], most recently in Refs. [7][8][9][10][11][12]. In this paper, we research the relation between helix-coil transition and folding.…”

Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram

“…The method also has been successfully applied to some complex systems, such as spin glass models [7,[14][15][16][17][18] and protein folding problems [8], for which the energy landscape is very rough and the conventional canonical Monte Carlo simulation gets easily trapped in local minima at low temperature. In the multicanonical method, we have to estimate the density of states g(E) first, then perform a random walk with a probability of 1 g(E) to make the histogram flat in the desired region in the phase space, such as between two peaks of the canonical distribution at the first-order transition temperature.…”

“…For example, cluster flip algorithms, beginning with the seminal work of Swendsen and Wang [3] and extended by Wolff [4], have been used to reduce critical slowing down near 2nd order transitions. Similarly, the multicanonical ensemble method [5] was introduced to overcome the tunneling barrier between coexisting phases at 1st order transitions and has general utility for systems with a rough energy landscape [6][7][8]. In both situations, histogram re-weighting techniques [9] can be applied in the analysis to increase the amount of information that can be gleaned from simulational data, but the applicability of re-weighting is severely limited in large systems by the statistical quality of the "wings" of the histogram.…”