Comparing a simple theoretical model for protein folding with all-atom molecular dynamics simulations

Abstract

Advances in computing have enabled microsecond all-atom molecular dynamics trajectories of protein folding that can be used to compare with and test critical assumptions of theoretical models. We show that recent simulations by the Shaw group (10,11,14,15) are consistent with a key assumption of an Ising-like theoretical model that native structure grows in only a few regions of the amino acid sequence as folding progresses. The distribution of mechanisms predicted by simulating the master equation of this native-centric model for the benchmark villin subdomain, with only two adjustable thermodynamic parameters and one temperature-dependent kinetic parameter, is remarkably similar to the distribution in the molecular dynamics trajectories. T heoretical models are essential for understanding the complex self-assembly reaction of protein folding, despite their reliance on apparently strong simplifying assumptions (1-5). In fact, the validated assumptions in a successful model point to the important features in protein folding. However, these assumptions are generally difficult to either justify or falsify by experiment. Fortunately, we can now examine them directly, thanks to recent advances in computing that allow us to study folding kinetics and mechanisms with all-atom molecular dynamics (MD) simulations (6-13). In particular, through the work of the Shaw group (10,11,14,15), many folding and unfolding transitions have been observed in long equilibrium trajectories. Here, and in the accompanying paper by Best et al. (16), we compare the key assumptions of a simple Ising-like model with the results from the all-atom MD simulations of the Shaw group (10,11,14,15). We also compare the distribution of folding mechanisms predicted by the model and the MD simulations.Ideally, to capture the full complexity of the process, a theoretical model for protein folding should enumerate all relevant conformations and specify the transition rates between them. To be useful, the model should also quantify the observables in equilibrium and kinetics experiments, and, like any theoretical model, be testable by new experiments. Among the very large number of proposed models (see bibliography in SI Text), a simple Ising-like model based on the contact map of the folded structure (17, 18), with just two adjustable thermodynamic parameters and a third parameter to set the time scale for kinetics, is of particular interest because it is the only one thus far that enumerates conformations and quantitatively explains so many different kinds of experimental results (19)(20)(21)(22). The two key assumptions of the Ising-like model are that amino acid interactions absent in the native structure play an insignificant role in determining the folding mechanism (23-25) and that native structure grows in only a few regions of the amino acid sequence as folding progresses. In the accompanying paper by Best et al. (16), the MD simulations for nine proteins with clear two-state kinetics simulated by the Shaw group (10,11,14,15) are found to be con...

Folding pathway of a multidomain protein depends on its topology of domain connectivity

“…2. This many-body feature allows this simple model to adequately describe the free-energy surfaces (3,6,18,38,39) and kinetics (17,39,40) of many proteins. An additional advantage of using this model is that the analytical forms of partition functions, Z WSME ðMÞ and Z WSME ðM DLD ; M ABD Þ, can be exactly derived (41,42); thus, free-energy surfaces and other quantities are readily calculated from Z WSME to obtain a clear view of the folding process.…”

“…Therefore, emphasis on continuous regions that have the native-like configuration is an important feature of the WSME model and is also the reason why the model can describe the cooperativity among multiple residues in a suitable manner in single-domain proteins (17). However, with this emphasis on the many-residue effects, the WSME model may overestimate the cooperativity among residues in larger proteins.…”

“…For example, consider one domain, a discontinuous domain, consisting of residues 1 ≤ i ≤ N 1 and N 2 ≤ i ≤ N, and another domain, a continuous domain, consisting of residues N 1 < i < N 2 . Because there is a tendency that the continuous parts of the chain form "islands" of ordered structures (17,18) and that these continuous parts of a sequence can be the nuclei for folding, the discontinuous domain may not be an independent folding unit, but may depend on the continuous domain. In this paper, we theoretically analyze the problem of how a folding pathway is selected in multidomain proteins that have a discontinuous domain by using Escherichia coli dihydrofolate reductase (DHFR) as an example and compare it with its circular permutant that consists only of continuous domains.…”

Kinetic modulation of a disordered protein domain by phosphorylation

“…All-atom, explicit solvent micro to millisecond MD simulations [29][30][31] have in recent years significantly contributed to our understanding of disordered states of proteins 29,30,32,33 . Computational studies on KID protein using replica exchanged implicit solvent MD simulations 34 , and short all-atom MD simulations 35 showed that KID is largely unstructured and phosphorylation barely affects its helical propensity.…”

A simple theoretical model goes a long way in explaining complex behavior in protein folding

“…Eaton and coworkers used this expanded version of the model to analyze in great depth the equilibrium and folding kinetics of the ultrafast folding villin headpiece subdomain (16). They then compared the folding pathways predicted by the fitted model with the results from long-timescale, all-atom simulations performed by the Shaw group (17). The comparison showed that the folding pathways observed in atomistic simulations involved local nucleation of native structure on no more than two regions, followed by the growth or closure of a single loop.…”