# The Role of Conformational Entropy in the Determination of Structural-Kinetic Relationships for Helix-Coil Transitions

^{*}

## Abstract

**:**

## 1. Introduction

_{3}-NH

_{2}. We follow the strategy of our previous study, while expanding upon the models employed, which clarifies the impact of model representation or, more precisely, conformational entropy on the resulting structural-kinetic relationships. More specifically, by varying the energetic parameters of a given model type, we keep the conformational entropy approximately fixed and demonstrate that the global, long timescale kinetic properties (i.e., ratio of folding to unfolding timescales) are determined precisely by the average helical content of the ensemble. Comparison between two distinct model types demonstrates a shift in the timescale ratios due to the change in model representation. Furthermore, by adjusting the steric interactions of one model type, we provide clear evidence that the conformational entropy is the dominant contributor to this shift. In contrast, we find that more local, faster kinetic processes are consistently determined by structural features of the ensemble, regardless of the precise model representation.

## 2. Computational Methods

#### 2.1. Coarse-Grained (CG) Models

#### 2.1.1. Hybrid Gō (Hy-Gō)

#### 2.1.2. PLUM

- the side chain van der Waals radius is decreased to 90% of its original value [20].
- the hydrogen-bonding interaction strength is decreased to 94.5% of its original value [30].
- the hydrogren-bonding interaction strength is decreased to 90% of its original value.
- the side chain interaction interaction strength is decreased to 95% of its original value.

#### 2.2. Simulation Details

_{3}-NH

_{2}, which has been extensively characterized both computationally and experimentally [31] and is often employed as a reference system for force field optimization. Throughout the manuscript, we refer to the system simply as ${\left(\mathrm{AAQAA}\right)}_{3}$.

#### 2.2.1. Hy-Gō

#### 2.2.2. PLUM

#### 2.3. Lifson-Roig Models

#### 2.4. Markov State Models

## 3. Results and Discussion

_{3}-NH

_{2}peptide, which contains 15 peptide bonds. From the LR point of view, there are ${2}^{15}$ states, determined by enumerating the various sequences of helical (h) and coil (c) states along the peptide backbone. However, this representation is inappropriate for constructing MSMs from the simulation trajectories. Instead, we systematically determine coarser, representative microstates, which correspond to collections of the more detailed configurations of the system. See the Methods section for a detailed description of microstate determination. For reference, we employ data from previous analysis of the ${\left(\mathrm{AAQAA}\right)}_{3}$ system [31]. In particular, we characterize the ensembles generated by CG simulation models with respect to those obtained from NMR experiments and from simulations of the ff03* all-atom (AA) model.

#### 3.1. Properties of the ${\left(\mathrm{AAQAA}\right)}_{3}$ Helix-Coil Transition

#### 3.2. Validation of Structural-Kinetic Relationships for ${\left(\mathrm{AAQAA}\right)}_{3}$

#### 3.3. Thermodynamics and Transition Network Topology

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CG | coarse-grained |

MSM | Markov state model |

AA | all-atom |

LR | Lifson-Roig |

Hy-Gō | Hybrid Gō |

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**Figure 1.**A visualization of the Hy-Gō model representation and interactions for ${\left(\mathrm{AAQAA}\right)}_{3}$. (

**Left**) Illustration of a native contact between ${\mathrm{C}}_{\alpha}$ atoms and a generic contact between ${\mathrm{C}}_{\beta}$ atoms, along with the corresponding parameters, $\{{\u03f5}_{\mathrm{nc}},{\u03f5}_{\mathrm{db}},{\u03f5}_{\mathrm{hp}}\}$, associated with these interactions. (

**Right**) The top two panels present the interaction potentials for the Gō-type interactions as a function of the model parameters. In the top panel, ${\u03f5}_{\mathrm{db}}=0.5{\u03f5}_{\mathrm{nc}}$. The bottom panel presents the Weeks-Chandler-Andersen-like potentials employed to model sterics along the peptide backbone.

**Figure 2.**(

**a**) Representative implied timescale test. Each line represents a different characteristic timescale of the Markov state model as a function of lag time. The vertical dashed line denotes the lag time chosen in this case. (

**b**) Representative Chapman-Kolmogorov test. The “helix” state corresponds to the microstate for each model with the highest helical content (see Methods section for details). The blue solid curve presents the probability decay of the helix state, as determined directly from the simulation trajectory, while the red dashed curve presents the same quantity determined from the MSM. The transparent cyan and orange regions denote the error bars for each quantity.

**Figure 3.**Temperature dependence of the Lifson-Roig helix propagation parameter, w (

**a**), and the average fraction of helical segments, $\langle {f}_{\mathrm{h}}\rangle $ (

**b**). The curves presented are a linear fit of the raw data. (

**c**) The slope of $ln\langle {f}_{\mathrm{h}}\rangle (1/T)$, denoted $\Delta {f}_{\mathrm{h}}$, versus the enthalphy of helix extension (determined as the slope of $lnw(1/T)$). Data from the Hy-Gō models are denoted with circle markers and are colored according to their relative cooperativity (as determined by $\Delta {f}_{\mathrm{h}}$). The results from the PLUM models are colored in gray-scale according to $\Delta {H}_{\mathrm{hb}}$. The energetic reparametrizations of PLUM are indicated with square markers, while the PLUM-ent model is indicated with X markers. The experimental and all-atom results taken from Best and Hummer [31] are denoted with triangle magenta and dark blue markers, respectively. Note that the models are aligned by shifting the temperature such that all models achieve ${\langle {f}_{\mathrm{h}}\rangle}^{\mathrm{expt}}$ at ${T}^{\ast}$.

**Figure 4.**(

**ai**) Temperature dependence of the ratio of folding to unfolding timescales, ${t}_{\mathrm{fol}}/{t}_{\mathrm{unf}}$. (

**aii**) Slope of the linear fit of $ln({t}_{\mathrm{fol}}/{t}_{\mathrm{unf}})$ as a function of inverse temperature versus the enthalphy of helix extension, $\Delta {H}_{\mathrm{hb}}$ (determined as the slope of $lnw(1/T)$). (

**bi**) Ratio of waiting time in the helix state, ${t}_{\mathrm{w}}^{i}[\mathrm{h}\to \mathrm{c}]$, to waiting time in the coil state, ${t}_{\mathrm{w}}^{i}[\mathrm{c}\to \mathrm{h}]$, for residue position i as a function of the average fraction of helical residues, $\langle {N}_{\mathrm{h}}\rangle $, and the average fraction of lone helices, $\langle {N}_{\mathrm{l}}\rangle $, at ${T}^{\ast}$. The free parameter, ${c}_{l}$, was determined by a fit over all models to maximize the pearson correlation coefficient, R, between the plotted quantities. Here, i corresponds to the terminal ends of the peptide (not including capping groups). (

**bii**) Slope of the linear fit of $ln({t}_{\mathrm{w}}^{i}[\mathrm{h}\to \mathrm{c}]/{\mathrm{t}}_{\mathrm{w}}^{\mathrm{i}}[\mathrm{c}\to \mathrm{h}])$ as a function of inverse temperature versus $\Delta {H}_{\mathrm{hb}}$, the slope of the linear fit of $\langle {N}_{\mathrm{h}}\rangle $ as a function of inverse temperature, and the slope of the linear fit of $\langle {N}_{\mathrm{l}}\rangle $ as a function of inverse temperature. The free parameters, ${d}_{h}$ and ${d}_{l}$, were determined by a fit over all models to maximize the pearson correlation coefficient, R, between the plotted quantities. Data from the Hy-Gō models are denoted with circle markers and are colored according to their relative cooperativity (as determined by $\Delta {f}_{\mathrm{h}}$). The results from the PLUM models are colored in gray-scale according to $\Delta {H}_{\mathrm{hb}}$. The energetic reparametrizations of PLUM are indicated with square markers, while the PLUM-ent model is indicated with X markers.

**Figure 5.**Average conditional path entropy in the folding direction versus the enthalpy of helix extension, $\Delta {H}_{\mathrm{hb}}$ (determined as the slope of $lnw(1/T)$).

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**MDPI and ACS Style**

Rudzinski, J.F.; Bereau, T.
The Role of Conformational Entropy in the Determination of Structural-Kinetic Relationships for Helix-Coil Transitions. *Computation* **2018**, *6*, 21.
https://doi.org/10.3390/computation6010021

**AMA Style**

Rudzinski JF, Bereau T.
The Role of Conformational Entropy in the Determination of Structural-Kinetic Relationships for Helix-Coil Transitions. *Computation*. 2018; 6(1):21.
https://doi.org/10.3390/computation6010021

**Chicago/Turabian Style**

Rudzinski, Joseph F., and Tristan Bereau.
2018. "The Role of Conformational Entropy in the Determination of Structural-Kinetic Relationships for Helix-Coil Transitions" *Computation* 6, no. 1: 21.
https://doi.org/10.3390/computation6010021