Thermodynamics of Conformational Transitions in a Disordered Protein Backbone Model

Abstract

Conformational entropy is expected to contribute significantly to the thermodynamics of structural transitions in intrinsically disordered proteins or regions in response to protein/ligand binding, posttranslational modifications, and environmental changes. We calculated the backbone (dihedral) conformational entropy of oligoglycine (Gly_N), a protein backbone mimic and model intrinsically disordered region, as a function of chain length (N=3, 4, 5, 10, and 15) from simulations using three different approaches. The backbone conformational entropy scales linearly with chain length with a slope consistent with the entropy of folding of well-structured proteins. The entropic contributions of second-order dihedral correlations are predominantly through intraresidue ϕ-ψ pairs, suggesting that oligoglycine may be thermodynamically modeled as a system of independent glycine residues. We find the backbone conformational entropy to be largely independent of global structural parameters, like the end-to-end distance and radius of gyration. We introduce a framework referred to herein as "ensemble confinement" to estimate the loss (gain) of conformational free energy and its entropic component when individual residues are constrained to (released from) particular regions of the ϕ-ψ map. Quantitatively, we show that our protein backbone model resists ordering/folding with a significant, unfavorable ensemble confinement free energy because of the loss of a substantial portion of the absolute backbone entropy. Proteins can couple this free-energy reservoir to distal binding events as a regulatory mechanism to promote or suppress binding.

Figure 1

ϕ-ψ free-energy map. For illustrative purposes, a two-dimensional histogram of ϕ,ψ dihedral angles was constructed across all residues from an MD trajectory of ${\text{Gly}}_{15}$. The histogram was converted to a free-energy map relative to the most populated bin. Regions with values greater than 3.5 kcal/mol are colored white. The map is partitioned into six regions that are used to define a conformational state of oligoglycine. States 2, 3, 4, 5, and 6 correspond to the β, ${\text{ppII}}_R$, ${\text{ppII}}_L$, ${\alpha }_R$, and ${\alpha }_L$ regions, respectively. State 1 corresponds to a high-energy region (Hi).

Figure 2

Backbone conformational entropy scaling with oligoglycine chain length or number of glycine residues. Superscripts indicate the method used. For the MIE entropy estimates, subscripts denote the order of approximation or truncation strategy. $S_2^{\text{MIE},\text{fit}}$ and $S_3^{\text{MIE},\text{fit}}$ are taken as the asymptotes of hyperbolic fits of $S_2^{\text{MIE}}$ and $S_3^{\text{MIE}}$, respectively, as functions of time (see Materials and Methods for more details).

Figure 3

Trajectories of ${\text{Gly}}_{15}$ conformational entropy estimates with respect to simulation time. The dashed lines indicate the asymptotes, $S_2^{\text{MIE},\text{fit}}$ and $S_3^{\text{MIE},\text{fit}}$, of the hyperbolic fits of $S_2^{\text{MIE}}$ and $S_3^{\text{MIE}}$, respectively, and are colored accordingly.

Figure 4

MIE conformational entropy estimates of ${\text{Gly}}_{15}$ as a function of end-to-end distance (left) and radius of gyration (right). Implicit solvent simulations of ${\text{Gly}}_{15}$ were performed with the distance between terminal carbons constrained to eight different values with a harmonic bias potential. Trajectories were then partitioned by radius of gyration. The first-order MIE approximation is given as ${\sum }_{i=1}^{2N_{\text{res}}}{S}_1\left(q_i\right)$ (i.e., the sum over one-dimensional terms in Eq. 9). Probability distributions of end-to-end distance and radius of gyration from explicit solvent simulations of ${\text{Gly}}_{15}$ are shaded in gray.

Figure 5

(Left) ${\text{Gly}}_{15}$ confinement free energy as a function of conformational states (i). The vertical bar chart is a histogram of $\mathit{\Delta }{A}_i^{\text{oligo}}$ values. (Right) A fraction of residues falling within the six partitions of the ϕ-ψ map (see Fig. 1) are shown as a function of the average confinement free energy $\left(\overline{\mathit{\Delta }A}\right)$ taken over a sliding, nonoverlapping window of 100 conformational states, ordered by increasing values of $\mathit{\Delta }{A}_i^{\text{oligo}}$. A windowing average was taken to reduce noise and is different than the weighted average, ${\langle \mathit{\Delta }A\rangle }^{\text{oligo}}$, which is taken across all conformations.

Figure 6

Thermodynamic cycle of an idealized transcription factor composed of a DNA-binding domain and regulatory domain (RD), exhibiting disorder-mediated, negative allostery. Upon binding DNA, the RD undergoes an order-to-disorder transition that entropically promotes binding. A cofactor may bind to the RD and prevent the favorable increase in conformational entropy (bottom right), thus decreasing the binding affinity of the transcription factor for its target DNA sequence. To see this figure in color, go online.