Folding protein alpha-carbon chains into compact forms by Monte Carlo methods.

A method is presented for generating folded chains of specific amino acid sequences on a simple cubic lattice. Monte Carlo simulations are used to transform extended geometries of simplified alpha-carbon chains for eight small monomeric globular proteins into folded states. Permitted chain transitions are limited to a few types of moves, all restricted to occur on the lattice. Crude residue-residue potentials derived from statistical structure data are used to describe the energies for each conformer. The low resolution structures obtained by this procedure contain many of the correct gross features of the native folded architectures with respect to average residue energy per nonbonded contact, segment density, and location of surface loops and disulfide pairs. Rms deviations between these and the native X-ray structures and percentage of native long-range contacts found in these final folded structures are 7.6 +/- 0.7 A and 48 +/- 3%, respectively. This procedure can be useful for predicting approximate tertiary interactions from amino acid sequence.     

Protein folding mechanisms and the multidimensional folding funnel

An important idea that emerges from the energy landscape theory of protein folding is that subtle global features of the protein landscape can profoundly affect the apparent mechanism of folding. The relationship between various characteristic temperatures in the phase diagrams and landmarks in the folding funnel at fixed temperatures can be used to classify different folding behaviors. The one-dimensional picture of a folding funnel classifies folding kinetics into four basic scenarios, depending on the relative location of the thermodynamic barrier and the glass transition as a function of a single-order parameter. However, the folding mechanism may not always be quantitatively described by a single-order parameter. Several other order parameters, such as degree of secondary structure formation, collapse and topological order, are needed to establish the connection between minimalist models and proteins in the laboratory. In this article we describe a simple multidimensional funnel based on two-order parameters that measure the degree of collapse and topological order. The appearance of several different “mechanisms” is illustrated by analyzing lattice models with different potentials and sequences with different degrees of design. In most cases, the two-dimensional analysis leads to a classification of mechanisms totally in keeping with the one-dimensional scheme, but a topologically distinct scenario of fast folding with traps also emerges. The nature of traps depends on the relative location of the glass transition surface and the thermodynamic barrier in the multidimensional funnel. Proteins 32:136–158, 1998. © 1998 Wiley-Liss, Inc.     

A Dynamical Low-Rank Approach to the Chemical Master Equation

Stochastic reaction kinetics have increasingly been used to study cellular systems, with applications ranging from viral replication to gene regulatory networks and to signaling pathways. The underlying evolution equation, known as the chemical master equation (CME), can rarely be solved with traditional methods due to the huge number of degrees of freedom. We present a new approach to directly solve the CME by a dynamical low-rank approximation based on the Dirac–Frenkel–McLachlan variational principle. The new approach has the capability to substantially reduce the number of degrees of freedom, and to turn the CME into a computationally tractable problem. We illustrate the accuracy and efficiency of our methods in application to two examples of biological interest.     

A Dynamical Model of Lipoprotein Metabolism

We present a dynamical model of lipoprotein metabolism derived by combining a cascading process in the blood stream and cellular level regulatory dynamics. We analyse the existence and stability of equilibria and show that this low-dimensional, nonlinear model exhibits bistability between a low and a high cholesterol state. A sensitivity analysis indicates that the intracellular concentration of cholesterol is robust to parametric variations while the plasma cholesterol can vary widely. We show how the dynamical response to time-dependent inputs can be used to diagnose the state of the system. We also establish the connection between parameters in the system and medical and genetic conditions.     

Dynamical Phase Transitions Reveal Amyloid-like States on Protein Folding Landscapes

Developing an understanding of protein misfolding processes presents a crucial challenge for unlocking the mysteries of human disease. In this article, we present our observations of β-sheet-rich misfolded states on a number of protein dynamical landscapes investigated through molecular dynamics simulation and Markov state models. We employ a nonequilibrium statistical mechanical theory to identify the glassy states in a protein’s dynamics, and we discuss the nonnative, β-sheet-rich states that play a distinct role in the slowest dynamics within seven protein folding systems. We highlight the fundamental similarity between these states and the amyloid structures responsible for many neurodegenerative diseases, and we discuss potential consequences for mechanisms of protein aggregation and intermolecular amyloid formation.     

Dynamical Phase Transitions Reveal Amyloid-like States on Protein Folding Landscapes

Developing an understanding of protein misfolding processes presents a crucial challenge for unlocking the mysteries of human disease. In this article, we present our observations of β-sheet-rich misfolded states on a number of protein dynamical landscapes investigated through molecular dynamics simulation and Markov state models. We employ a nonequilibrium statistical mechanical theory to identify the glassy states in a protein’s dynamics, and we discuss the nonnative, β-sheet-rich states that play a distinct role in the slowest dynamics within seven protein folding systems. We highlight the fundamental similarity between these states and the amyloid structures responsible for many neurodegenerative diseases, and we discuss potential consequences for mechanisms of protein aggregation and intermolecular amyloid formation.     

Molecular Chaperones and Mitochondrial Protein Folding

Precursor proteins destined for the mitochondrial matrix traverse inner and outer organelle membranes in an extended conformation. Translocation events are therefore integrally coupled to the processes of protein unfolding in the cytosol and protein refolding in the matrix. To successfully import proteins from the cytoplasm into mitochondria, cells have recruited a variety of molecular chaperone systems and folding catalysts. Within the organelles, mitochondrial Hsp70 (mt-Hsp70) is a major player in this process and exerts multiple functions. First, mt-Hsp70 binds together with cohort proteins to incoming polypeptide chains, thus conferring unidirectionality on the translocation process, and then assists in their refolding. A subset of imported proteins requires additional assistance by chaperonins of the Hsp60/Hsp10 family. Protein folding occurs within the cavity of these cylindrical complexes. A productive interaction of precursor proteins with molecular chaperones in the matrix is not only crucial for correct refolding and assembly, but also for processing of presequences, intramitochondrial sorting, and degradation of proteins. This review focuses on the role of mt-Hsp70 and Hsp60/Hsp10 in protein folding in the mitochondrial matrix and discusses recent findings on their molecular mechanism of action.     

Lattice simulations of cotranslational folding of single domain proteins

We use lattice protein models and Monte Carlo simulations to study cotranslational folding of small single domain proteins. We show that the assembly of native structure begins during late extrusion stages, but final formation of native state occurs during de novo folding, when all residues are extruded. There are three main results in our study. First, for the sequences displaying two-state refolding mechanism de novo cotranslational folding pathway differs from that sampled in in vitro refolding. The change in folding pathways is due to partial assembly of native interactions during extrusion that results in different starting conditions for in vitro refolding and for de novo cotranslational folding. For small single domain proteins cotranslational folding is slower than in vitro refolding, but is generally fast enough to be completed before the release from a ribosome. Second, we found that until final stages of biosynthesis cotranslational folding is essentially equilibrium. This observation is explained by low stability of structured states for partially extruded chains. Finally, our data suggest that the proteins, which refold in vitro slowly via intermediates, complete their de novo folding after the release from a ribosome. Comparison of our lattice cotranslational simulations with recent experimental and computational studies is discussed.     

蛋白质的折叠

重点介绍了蛋白质折叠的热力学控制学说和动力学控制学说,简单介绍了几种蛋白质折叠模型并分析了多肽链在体内进行快速折叠的原因。     

The Protein Folding Problem: The Role of Theory

The protein folding problem was first articulated as question of how order arose from disorder in proteins: How did the various native structures of proteins arise from interatomic driving forces encoded within their amino acid sequences, and how did they fold so fast? These matters have now been largely resolved by theory and statistical mechanics combined with experiments. There are general principles. Chain randomness is overcome by solvation-based codes. And in the needle-in-a-haystack metaphor, native states are found efficiently because protein haystacks (conformational ensembles) are funnel-shaped. Order-disorder theory has now grown to encompass a large swath of protein physical science across biology.     

一种在格点模型上模拟蛋白质折叠结构的优化算法

蛋白质折叠问题是生物信息学中一个经典的多项式复杂程度的非确定性(non-deterministic polynomial,NP)难度问题.势能曲面变平法(ELP)是一种启发式的全局优化算法.通过对ELP方法中的直方图函数提出一种新的更新机制,并将基于贪心策略的初始构象的产生,基于牵引移动的邻域搜索策略与ELP方法相结合,为面心立方体(FCC)格点模型的蛋白质折叠问题提出一种改进的势能曲面变平(ELP+)算法.采用文献中9条常用序列作为测试集.对于每条序列,ELP+算法均能找到与文献中的算法所得到的最低能量相等或更低的能量.实验结果表明,ELP+算法是求解FCC格点模型的蛋白质折叠问题的一种有效算法.     

Folding simulations and computer redesign of protein A three-helix bundle motifs

In solution, the B domain of protein A from Staphylococcus aureus (B domain) possesses a three-helix bundle structure. This simple motif has been previously reproduced by Kolinski and Skolnick (Proteins 18: 353–366, 1994) using a reduced representation lattice model of proteins with a statistical interaction scheme. In this paper, an improved version of the potential has been used, and the robustness of this result has been tested by folding from the random state a set of three-helix bundle proteins that are highly homologous to the B domain of protein A. Furthermore, an attempt to redesign the B domain native structure to its topological mirror image fold has been made by multiple mutations of the hydrophobic core and the turn region between helices I and II. A sieve method for scanning a large set of mutations to search for this desired property has been proposed. It has been shown that mutations of native B domain hydrophobic core do not introduce significant changes in the protein motif. Mutations in the turn region were also very conservative; nevertheless, a few mutants acquired the desired topological mirror image motif. A set of all atom models of the most probable mutant was reconstructed from the reduced models and refined using a molecular dynamics algorithm in the presence of water. The packing of all atom structures obtained corroborates the lattice model results. We conclude that the change in the handedness of the turn induced by the mutations, augmented by the repacking of hydrophobic core and the additional burial of the second helix N-cap side chain, are responsible for the predicted preferential adoption of the mirror image structure. © 1996 Wiley-Liss, Inc.     

Back to Units of Protein Folding

Abstract

In response to the criticism by A. Finkelstein (J. Biomol. Struct. Dyn. 20, 311–314, 2002) of our Communication (J. Biomol. Struct. Dyn. 20, 5–6, 2002) several issues are dealt with. Importance of the notion of elementary folding unit, its size and structure, and the necessity of further characterization of the units for the elucidation of the protein folding in vivo are discussed. The criticism (J. Biomol. Struct. Dyn. 20, 311–314, 2002) on the hierarchical protein folding is also briefly addressed.     

Design and Structure of an Equilibrium Protein Folding Intermediate: A Hint into Dynamical Regions of Proteins

Partly unfolded protein conformations close to the native state may play important roles in protein function and in protein misfolding. Structural analyses of such conformations which are essential for their fully physicochemical understanding are complicated by their characteristic low populations at equilibrium. We stabilize here with a single mutation the equilibrium intermediate of apoflavodoxin thermal unfolding and determine its solution structure by NMR. It consists of a large native region identical with that observed in the X-ray structure of the wild-type protein plus an unfolded region. Small-angle X-ray scattering analysis indicates that the calculated ensemble of structures is consistent with the actual degree of expansion of the intermediate. The unfolded region encompasses discontinuous sequence segments that cluster in the 3D structure of the native protein forming the FMN cofactor binding loops and the binding site of a variety of partner proteins. Analysis of the apoflavodoxin inner interfaces reveals that those becoming destabilized in the intermediate are more polar than other inner interfaces of the protein. Natively folded proteins contain hydrophobic cores formed by the packing of hydrophobic surfaces, while natively unfolded proteins are rich in polar residues. The structure of the apoflavodoxin thermal intermediate suggests that the regions of natively folded proteins that are easily responsive to thermal activation may contain cores of intermediate hydrophobicity.     

Stoichiometry-driven Protein Folding: A Comment

Abstract

We have found, with the aid of 2-D gel electrophoresis, that double-stranded human telomeric repeat, (T₂AG₃)₁₂·(C₃TA₂)₁₂, being cloned within a plasmid, forms a protonated superhelically-induced structure. Experiments on chemical and enzymatic probing also indicate that the human telomeric repeats adopt an unusual structure. We have proposed an eclectic model for this structure in which four different elements coexist: a non-orthodox intramolecular triplex stabilized by the canonical protonated C · G*C⁺ base-triads and highly enriched by non-canonical base-triads; the intramolecular quadruplex formed by a portion of the G-rich strand; the single-stranded region encompassing a portion of the G-rich strand and, probably, the (C,A)-hairpin formed by a portion of the C-rich strand.     

Pathway Diversity and Concertedness in Protein Folding: An ab-initio Approach

Abstract

Making use of an ab-initio folding simulator, we generate in vitro pathways leading to the native fold in moderate size single-domain proteins. The assessment of pathway diversity is not biased by any a-priori information on the native fold. We focus on two study cases, hyperthermophile variant of protein G domain (1gb4) and ubiquitin (1ubi), with the same topology but different context dependence in their native folds. We demonstrate that a quenching of structural fluctuations is achieved once the proteins find a stationary plateau maximizing the number of highly protected hydrogen bonds. This enables us to identify the folding nucleus and show that folding does not become expeditious until a concerted event takes place generating a topology able to prevent water attack on a maximal number of hydrogen bonds. This result is consistent with the standard nucleation mechanism postulated for two-state folders. Pathway diversity is correlated with the extent of conflict between local structural propensity and large-scale context, rather than with contact order: In highly context-dependent proteins, the success of folding cannot rely on a single fortuitous event in which local propensity is overruled by large-scale effects. We predict mutational Φ values on individual pathways, compute ensemble averages and predict extent of surface burial and percentage of hydrogen bonding on each component of the transition state ensemble, thus deconvoluting individual folding-route contributions to the averaged two-state kinetic picture. Our predicted kinetic isotopic effects find experimental support and lead to further probes. Finally, the molecular redesign potentiality of the method, aimed at increasing folding expediency, is explored.     

Chaperonin-mediated Protein Folding

We have been studying chaperonins these past twenty years through an initial discovery of an action in protein folding, analysis of structure, and elucidation of mechanism. Some of the highlights of these studies were presented recently upon sharing the honor of the 2013 Herbert Tabor Award with my early collaborator, Ulrich Hartl, at the annual meeting of the American Society for Biochemistry and Molecular Biology in Boston. Here, some of the major findings are recounted, particularly recognizing my collaborators, describing how I met them and how our great times together propelled our thinking and experiments.     

Dynamical phase coexistence: A simple solution to the “savanna problem”

We introduce the concept of dynamical phase coexistence to provide a simple solution for a long-standing problem in theoretical ecology, the so-called “savanna problem”. The challenge is to understand why in savanna ecosystems trees and grasses coexist in a robust way with large spatiotemporal variability. We propose a simple model, a variant of the contact process (CP), which includes two key extra features: varying external (environmental/rainfall) conditions and tree age. The system fluctuates locally between a woodland and a grassland phase, corresponding to the active and absorbing phases of the underlying pure contact process. This leads to a highly variable stable phase characterized by patches of the woodland and grassland phases coexisting dynamically. We show that the mean time to tree extinction under this model increases as a power-law of system size and can be of the order of 10,000,000 years in even moderately sized savannas. Finally, we demonstrate that while local interactions among trees may influence tree spatial distribution and the order of the transition between woodland and grassland phases, they do not affect dynamical coexistence. We expect dynamical coexistence to be relevant in other contexts in physics, biology or the social sciences.