We consider a general incompressible finite model protein of size M in its environment, which we represent by a semiflexible copolymer consisting of amino acid residues classified into only two species (H and P, see text) following Lau and Dill. We allow various interactions between chemically unbonded residues in a given sequence and the solvent (water), and exactly enumerate the number of conformations W(E) as a function of the energy E on an infinite lattice under two different conditions: (i) we allow conformations that are restricted to be compact (known as Hamilton walk conformations), and (ii) we allow unrestricted conformations that can also be non-compact. It is easily demonstrated using plausible arguments that our model does not possess any energy gap even though it is supposed to exhibit a sharp folding transition in the thermodynamic limit. The enumeration allows us to investigate exactly the effects of energetics on the native state(s), and the effect of small size on protein thermodynamics and, in particular, on the differences between the microcanonical and canonical ensembles. We find that the canonical entropy is much larger than the microcanonical entropy for finite systems. We investigate the property of self-averaging and conclude that small proteins do not self-average. We also present results that (i) provide some understanding of the energy landscape, and (ii) shed light on the free energy landscape at different temperatures.
Proteins are organic compounds made of amino acids, also known as residues, bound in a chain-like structure by peptide bonds. Self-assembling small proteins can fold into their native states (of minimum free energy) without any chaperones, and have been extensively investigated recently using lattice models by thermodynamic principles [1]. They differ from flexible polymers, which collapse to a compact disordered state; they are similar to semiflexible polymers in which semiflexibility forces an ordered (crystalline) compact structure at low temperatures [2].
Let N R denote the total number of residues in N proteins in a volume V ; the residue concentration is
To ensure that the boundary of the volume V does not affect the behavior of the system, we need to take the limit V → ∞. This limit will be usually implicit in the following, unless mentioned otherwise. In many cases, we deal with a dilute solution so that the concentration of proteins is exceedingly small. Accordingly, the proteins are far apart with no appreciable inter-protein interactions. It is then safe to consider a single protein by itself in its environment, i.e. in the presence of water. The presence of inter-protein interactions in a solution, which is not dilute, and in a bulk means that these systems (both of which we will not consider in this work) containing many proteins should be distinguished from that containing a single protein, as their thermodynamics will be very different.
Our focus in this work is on a single protein (N = 1) containing M residues so that N R = M . As proteins are usually small in size, we need to recognize that the behavior of a single protein is governed by the thermodynamics of a small system (defined as a system in which N R does not grow with the volume V as V → ∞) and not of a macroscopic system, such as formed by a bulk (in which N R ≡ N M grows with the volume V ); the latter will be governed by the thermodynamics of a macroscopic system [3]. It is well known that predictions of different ensembles describing a macroscopic system are the same, except at some singular points such as where phase transitions occur. Therefore, it is important to understand the ways in which different statistical ensembles differ from each other for small systems. This is one of the important issues motivating this investigation: how to distinguish small system thermodynamics from a macroscopic system thermodynamics in various ensembles. For this purpose, it is sufficient to consider only two ensembles: the microcanonical (ME) and the canonical (CE) ensembles.
The residue sequence (known as the primary structure) in a protein is defined by a gene and is encoded in the corresponding genetic code. Understanding the relationship between the sequence and protein functionality is an unsolved problem though major progress has been made [4]. A first-principle study of primary, secondary (regularly repeating local structures, such as helices and βsheets) and tertiary (the overall shape or conformations of a single protein) structures requires short (local) and long (nonlocal) ranged model energetics that, while remaining independent of protein conformations, temperature and pressure, determines the native state(s), and has to be judiciously chosen to give a unique and correct native state [5].
The simplest model that can be used is the standard model of Lau and Dill [6], which classifies the 20 different amino acid groups or residues into two subsets, H (hydrophobic residues) and P (hydrophilic/polar residues), and allows only nearest-neighbor attractive HH interaction (whose strength is set equal to 1 in some predetermined unit) to provide good hydrophobic cores; however, consideration of local energetics of the 20 residues [7] is also common. It is also found that the introduction of multi-body interaction enhances cooperativity [8], and should not be neglected.
The protein in the standard model is an example of a copolymer of a prescribed sequence. It is this simplified copolymer model and its variants proposed in this work that will be the subject of investigation here, even though the work can be extended to a more general case.
The microscopic energies that appear in the model energetics, while determining the thermodynamics, must themselves be independent of the thermodynamic state, i.e., of protein conformations, temperature, pressure, concentration, etc. to be truly microscopic. In addition, a proper model should satisfy certain principles [9], one of which is the requirement of cooperativity needed for the existence of a first-order transition (a latent heat) at the folding transition to the native state. The residue sequence plays an important role in determining the native state [10] and, therefore, the thermodynamics. Thus, we are driven to treat proteins as semiflexible heteropolymers with certain specific sequences [11]. However, there is no consensus for general energetics to describe all proteins, and there re
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