Role of single-point mutations and deletions on transition temperatures in ideal proteinogenic heteropolymer chains in the gas phase

Abstract

A coarse-grained statistical mechanics-based model for ideal heteropolymer proteinogenic chains of non-interacting residues is presented in terms of the size K of the chain and the set of helical propensities $\Omega = \{ \omega _j \}$ associated with each residue j along the chain. For this model, we provide an algorithm to compute the degeneracy tensor $G_{K}^{\alpha }$ associated with energy level $E_{\alpha }$ where $\alpha$ is the number of residues with a native contact in a given conformation. From these results, we calculate the equilibrium partition function $Z_{het}$ and characteristic temperature $T_c$ at which a transition from a low to a high entropy states is observed. The formalism is applied to analyze the effect on characteristic temperatures $T_c$ of single-point mutations and deletions of specific amino acids $X_j$ along the chain. Two probe systems are considered. First, we address the case of a random heteropolymer of size K and given helical propensities $\Omega$ on a conformational phase space. Second, we focus our attention to a particular set of neuropentapeptides, [Met-5] and [Leu-5] enkephalins whose thermodynamic stability is a key feature on their coupling to $\delta$ and $\kappa$ receptors and the triggering of biochemical responses.

Appendix: Transformation from microscopic variables \((U, \epsilon , \Upsilon )\) from ZW homopolymer model to macroscopic variables \((f, \Delta , \Omega )\) of heteropolymer model

It is possible to rewrite the ZW homopolymer model with original variables $(U, \epsilon , \nu )$ in terms of new triplet $(f, \Delta , \omega )$ Olivares-Quiroz (2013). Variables f and $\Delta$ are close related to the structure of the energy spectrum proposed in Eq. (1). Variable f is defined

$$\begin{aligned} f= \left| \frac{\epsilon }{KU} \right| >0 \end{aligned}$$

(8)

which is a dimensionless quantity and represents a measure of the energy scales involved. Depending of the value of f, three different regimes are embedded in the energy spectrum. If f is much smaller than unity, i.e., $f \ll 1$, then the energy KU of the unfolded structure is much larger than the energy $\epsilon$ of the native state and then the transition to the native state is poorly favored. On the other hand, if $f \gg 1$, the energy of the native state is much larger than the energy of the unfolded structure, a disorder-order transition is favored. For intermediate values $f \simeq$, both unfolded and native states are equally favored. Variable $\Delta$ is defined in terms of the energy gap between the energies of the complete disordered and ordered structures. Then, $\Delta$ as

$$\begin{aligned} \Delta = -(KU + \epsilon ) \end{aligned}$$

(9)

with $KU >0$ and $\epsilon >0$ positive quantities. Variable $\Delta$ measures the energy difference between the complete unfolded and folded conformations. It can be compared directly with experimental measurements since it is usually the energy gap between unfolded and folded structures in protein and peptides what is measured. Standard measurements assign a value of $\Delta = [-50,-60]$ kJ mol$^{-1}$ for most unfolding transitions.

It is possible to establish a connection between variable $\nu _j$ and the helical propensity $\omega _j$ of particular residue $X_j$ which is a quantitative measurement of how prone it is to participate in a helical conformation. According to Lifson-Roig theory for helix-coil transition $\omega$ is defined as a configurational integral over the region of the space-phase of dihedral angles $(\phi , \psi )$ that corresponds to a helical conformation of the interacting potential $V(\phi , \psi )$ of the chain (Lifson 1961; Scheraga 1970). In addition, experimental measurements of helical propensity of each of the 20 proteinogenic residues as a function of position and temperature have been achieved recently (Doig and Baldwin 1995; Scholtz et al. 1991; Petukhov et al. 1998; Chakrabartty et al. 1994; Creamer and Rose 1992). Of particular interest here is the scale proposed by Chakrabartty et al. (1994), which assigns to amino acid alanine (Ala) the highest helical propensity and views the other amino acid residues as “helix-breakers” of varying strength.

In order to connect $\omega _j$ to the number of potential non-native conformations $\nu _j$ per residue, it is important to recall that from its definition it can be considered as both a conditional probability and a configurational partition function per residue between helical and non-helical conformations. In this sense, the change in free energy per residue $\Delta \hat{F}$ can be written as $\Delta \hat{F}= \Delta \hat{E} - T \Delta \hat{S}$, where $\Delta \hat{E}$ is the change in internal energy per residue during conformational transitions and $\Delta \hat{S}$ is the corresponding entropy change per residue as well. Particularly, the entropy change $\Delta \hat{S}$ can be understood in terms of the change in the number of conformations available to each residue. Then the entropy change $\Delta \hat{S}$ due to conformational transition from non-native to native state per residue is

$$\begin{aligned} \Delta \hat{S}= R \ln {(1)- R \ln {(\nu )}}= R \ln {(1/\nu )}=- R \ln {\nu }, \end{aligned}$$

(10)

where $\nu >1$ always. Let us underscore that since $\nu >1$ thus the entropy change $\Delta \hat{S}$ is always negative as it corresponds from a pure conformational transition from multiple microstates to a single state. Then, inserting Eq. (10) into the expression for the free energy $\Delta \hat{F}$ given above, we obtain

$$\begin{aligned} \omega = \frac{1}{\nu } \exp { \left( \frac{-\Delta E}{RT} \right) } \end{aligned}$$

(11)

which establishes an inversely proportional relationship between $\omega$ and $\nu$. Equation (11) also has the functional form of an activation energy barrier between the ordered and non-ordered conformations of the same type as an Arrhenius law.

Using Eq. (11), it is then possible to establish a connection between the set $\Upsilon =\{ \nu _1, \nu _2, \ldots \nu _K \}$ of the model and the set $\Omega = \{ \omega _1, \omega _2, \ldots \omega _K\}$ of experimental helical propensities. From this, the heteropolymer partition function $Z_{het}$ can be written succinctly as

$$\begin{aligned} Z_{het} (\beta , f, \Delta , \Omega , K)= \sum _{\alpha =0}^{K} G_{\alpha }^{K} (\Omega , K)\exp \left[ -\beta E_{\alpha } (f, \Delta )\right] \end{aligned}$$

(12)

where the degeneracy $G_{\alpha }^{K}$ is a explicit function of the set $\Omega =\{ \omega _1, \omega _2, \ldots \omega _K \}$ and the size K of the chain. The energy spectrum $E_{\alpha }$ contains the dependence of penalty energies U through new variables f and $\Delta.$ In Eq. (12) it is assumed that all residues have the same penalty energy U for a non-native contact.