During the synthesis of integral membrane proteins (IMPs), the hydrophobic amino acids of the polypeptide sequence are partitioned mostly into the membrane interior and hydrophilic amino acids mostly into the aqueous exterior. We analyze the minimum free energy state of polypeptide sequences partitioned into alpha-helical transmembrane (TM) segments and the role of thermal fluctuations using a many-body statistical mechanics model. Results suggest that IMP TM segment partitioning shares important features with general theories of protein folding. For random polypeptide sequences, the minimum free energy state at room temperature is characterized by fluctuations in the number of TM segments with very long relaxation times. Simple assembly scenarios do not produce a unique number of TM segments and jamming phenomena interfere with segment placement. For sequences corresponding to IMPs, the minimum free energy structure with the wildtype number of segments is free of number fluctuations due to an anomalous gap in the energy spectrum, and simple assembly scenarios produce this structure. There is a threshold number of random point mutations beyond which the size of this gap is reduced so that the wildtype groundstate is destabilized and number fluctuations reappear.
Deep Dive into Statistical Mechanics of Integral Membrane Protein Assembly.
During the synthesis of integral membrane proteins (IMPs), the hydrophobic amino acids of the polypeptide sequence are partitioned mostly into the membrane interior and hydrophilic amino acids mostly into the aqueous exterior. We analyze the minimum free energy state of polypeptide sequences partitioned into alpha-helical transmembrane (TM) segments and the role of thermal fluctuations using a many-body statistical mechanics model. Results suggest that IMP TM segment partitioning shares important features with general theories of protein folding. For random polypeptide sequences, the minimum free energy state at room temperature is characterized by fluctuations in the number of TM segments with very long relaxation times. Simple assembly scenarios do not produce a unique number of TM segments and jamming phenomena interfere with segment placement. For sequences corresponding to IMPs, the minimum free energy structure with the wildtype number of segments is free of number fluctuations d
Anfinsen established in a landmark study that the three-dimensional structure of globular proteins is determined by their primary amino acid sequences and that this structure is a minimum free energy state [1]. Integral membrane proteins (IMPs) such as ion channels, ion pumps, porins, and receptor proteins, do not easily lend themselves to Anfinsen's method, and whether or not assembled IMPs represent global free energy minima is not known [2]. The focus of this paper is on one of the most common IMP structures: bundles of, typically, seven to twelve transmembrane (TM) α-helices Only the structures produced by sequential assembly with P = P w reduce, at room temperature, to a minimum free energy state.
In the presence of point mutations, the stabilizing anomalous energy gap shrinks as the number of random point mutations increases until a threshold is reached marked by growth of number fluctuations.
The contrast between the wildtype and random sequences in terms of thermodynamics and assembly kinetics is rather similar to that between the glassy “molten globule” state of collapsed generic polypeptide sequences in bulk solutions and the “designed” folded state of globular proteins at the lowest point of the folding “funnel” [11]. This folded state is usually free of large-scale, destabilizing thermal fluctuations, and is accessible from the unfolded state by rapid assembly kinetics.
This suggests that IMPs and globular proteins can be described by a common general phenomenology.
The
with δ (j) the hydrophobicity of residue j, for which we use the scale of Reference [9]. no free energy penalty. Specifically, for a rod of species α starting at site j followed by another rod (of any species) starting at site k > j , the interaction potential is
It should be noted that the aim of this model is
The term ! " F (k) represents the “forward” Boltzmann statistical weight of all possible TM segment distributions located anywhere between sites 1 and k given that that there is a TM segment of size L ! that starts at site
defined as the probability that a residue k is part of a TM segment of any allowed size. A plot of ! (k) † In the grand canonical ensemble, it corresponds to the second derivative of the thermodynamic potential with respect to the chemical potential.
shows the most probable locations of the TM segments.
Mathematically, the problem of computing TM placement probabilities has now been reduced to the computation of the grand canonical partition function ! and the site-specific “one-sided” partition functions
multi-species liquid of variable-sized hard rods subject to an external potential. In the transfer matrix method, one first breaks up ! " F (k) as a sum over the different possible values of the distance k ! j (in residues) between a segment of size L ! starting at k and a neighboring segment starting at site j with
The term W ! ," (k ! " F (k) for k > 1 can be computed by forward iteration. A similar relation holds for the backward weights:
which is reconstructed starting from ! " B (N ) = 1 . Using these recursion relations it is possible to numerically reconstruct ! µ ( ) under conditions of thermodynamic equilibrium for any given amino acid sequence. The transfer matrix method is closely related to hidden Markov models [7] while for the case that all segments have the same size, it reduces to the analytically soluble Percus model [12] of hard rods in an external potential.
Fluctuations. In order to interpret occupancy profiles of this form, it is useful to overlay them on the hydropathy plot, as is done in !G " (k) can be located for the given µ. All four rows place the segments in approximately the same locations. Figure 3A shows the mean segment number ! SA (µ)
obtained by sequential adsorption for the bR sequence as a dashed line. Sequential adsorption exactly reproduces ! TM (µ) up to and including P = 7 but then sequential adsorption halts while ! TM (µ) continues to increase. This “jamming” phenomenon is a familiar feature of studies of sequential adsorption in other systems [14]. The case of and ! TM (µ) for cytochrome C oxidase and diacylglycerol kinase).
Recall that we found that the room temperature susceptibility χ for number fluctuations was negligible for P = P w at the center of the wildtype stability interval, so number fluctuations were not required for shown in column 1 with their corresponding PDB id. The third column gives the size of the stability interval Δµ for P = P w of the groundstate structure as computed from the model. The fourth column gives the average (over 100 runs) number of random single point mutations (SPMs) normalized by sequence length required to change the segment from its wildtype value for each protein, including standard error.
Note that 1PW4 has the maximum thermodynamic stability Δµ but a low mutation threshold. The fifth column gives the average (over 100 runs) mutation threshold for each sequence after random shuffling (10
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