Conformation 101 - Funneled energy landscape

Thoughts on protein conformation change.

Systematic descriptions of the energy landscapes help us to intrinsically quantify conformations and their changes. Here I will give a brief summary of pioneered works of Prof. Jin Wang et al. introducing the funneled energy landscape of protein structures.

For analytical derivation, we can assume the energy $E$ of a conformation (of a protein sequence) to be the sum of independent interactions, thus the density of states (i.e. a statistical energy distribution in microcanonical ensemble) approximates a Gaussian distributionNoted that the density of states can be obtained from the simulated canonical ensemble after transformation and is not necessarily follows Gaussian distribution in a real system.:

\[n(E) = \frac{1}{\sqrt{2\pi\Delta E^2}}\exp\left[-\frac{(E-\bar{E})^2}{2\Delta E^2}\right],\quad\Delta E = \sqrt{\langle E^2 \rangle - \langle E \rangle^2}\]

For a protein sequence, its total number of conformations is assumed to be $\Omega_0$. Then the conformational entropy with energy $E$ is:

\[\begin{aligned} S(E) &= K_B\ln(\underbrace{\Omega_{0} n(E)}_{\text{the number of conformations with energy} E}) \\ &= K_B \ln \Omega_0 + K_B \ln n(E)\\ &= \underbrace{K_B \ln \Omega_0}_{S_0} - K_B \frac{(E-\bar{E})^2}{2\Delta E^2} \color{gray}{\underbrace{- K_B \frac{\ln(2\pi\Delta E^2)}{2}}_{\text{dropped}}} \end{aligned}\]

Since we have the thermodynamic relation:

\[\frac{\partial S}{\partial E} = \frac{1}{T}\]

we can derive the most probable energy as a function of $T$ as:

\[\begin{aligned} \frac{-2K_{B}(E-\bar{E})}{2 \Delta E^{2}}&= \frac{1}{T}\\ &\Downarrow \\ E(T) &= \bar{E} - \frac{\Delta E^{2}}{K_{B}T} \\ \end{aligned}\]

So the entropy at the most probable energy as a function of $T$ is:

\[S(T) = S(E(T)) = S_{0} - \frac{\Delta E^{2}}{2K_{B}T^{2}}\]

• (noted that) for an infinite high temperature, $S(T)$ is approximately $S_{0}$.
• (noted that) when at or below a lower enough critical temperature $T_g$ ($S(T_g)=0$), the protein is trapped in one of the frozen states.

\[T_g = \sqrt{\frac{\Delta E^{2}}{2 K_B S_0}}\]

From the thermodynamic expressions of the energy $E(T)$ and entropy $S(T)$, the Helmholtz free energy of the system as a function of $T$ can be expressed as:

\[\begin{aligned} F(T) &= E(T) - TS(T)\\ &=\bar{E} - \frac{\Delta E^{2}}{K_{B}T} - TS_{0} + \frac{\Delta E^{2}}{2K_{B}T}\\ &= \bar{E} - TS_{0} - \frac{\Delta E^{2}}{2K_{B}T}\\ \end{aligned}\]

With the observations of naturally occurring proteins, we can assume that a natural protein normally has a unique ground state (with energy $E_N$) at which both the energy and entropy variance (roughness) are zero. Then the free energy of the native state equals $E_N$.

A first-order transition between native state and non-native state is expected at the temperature $T_{f}$ (folding transition temperature) where they have equal free energy. Thus we have:

\[\begin{aligned} E_{N} &= \bar{E} - T_{f}S_{0} - \frac{\Delta E^{2}}{2K_{B}T_{f}} \\ \underbrace{\bar{E}-E_{N}}_{\delta E} &= T_{f}S_{0} + \frac{\Delta E^{2}}{2K_{B}T_{f}} \\ &\Downarrow \\ T_{f} &= \frac{\delta E}{2S_{0}}(1+\sqrt{1 - \frac{2S_{0}\Delta E^{2}}{K_{B} \delta E^{2}}}) \end{aligned}\]

$T_{f}$ should be larger than $T_{g}$ so as not to be trapped in the frozen state and make the system able to reach the native state. So we are focusing on $\frac{T_{f}}{T_{g}}$ and find out that:

\[\begin{aligned} \frac{T_{f}}{T_{g}} &= \underbrace{\sqrt{\frac{K_{B}}{2S_{0}}}\frac{\delta E}{\Delta E}}_{\Lambda} + \sqrt{\frac{K_{B}\delta E^{2}}{2S_{0}\Delta E^{2}}-1} \\ &= \Lambda + \sqrt{\Lambda^2 - 1} \\ \end{aligned}\]

• The larger the ratio $\frac{T_{f}}{T_{g}}$ is, the less chance the protein has to be trapped on the way to its native state.
• $\Lambda$ itself is a quantitative measure of the landscape topogarphy. The larger the $\Lambda$ is, the more funneled protein folding energy landscape shape is against the vast number of states and roughness (i.e. minimal frustration principle).
• Maximizing $\frac{T_{f}}{T_{g}}$ is equivalent of maximizing the value of $\Lambda$.
  • This relationship connects the folding criterion to the underlying landscape topography.
  • It provides a practical implementation of the principle of minimal frustration for protein folding.
  • Optimization of $\Lambda$ guarantees the kinetic accessibility of the native state.

So we have the Optimal Foldability Criterion:

\[\max \frac{T_f}{T_g} \sim \max \Lambda\]

The probability $P(E)$ of sampling any conformation at a finite temperature $T$ with energy $E$ is:

\[P(E) = \frac{n(E)\exp[-\frac{E-\bar{E}}{K_{B}T}]}{Z}\]

• Transforming from the microcanonical ensemble to the canonical ensemble, the probability is weighted by the Boltzmann factor $\exp[-\frac{E-\bar{E}}{K_{B}T}]$.
• $Z$ is the partition function of the canonical ensemble.

Thus, at a particular $T$ larger than $T_{g}$, the probability of the system in its unique native state and the non-native state are $P_N$ and $P_{D}$ respectively:

\[\begin{aligned} P_{N}&=\frac{\exp[-\frac{E_{N}-\bar{E}}{K_{B}T}]}{Z}\\ P_{D}&=\sum_{E>E_{N}}\frac{n(E)\exp[-\frac{E-\bar{E}}{K_{B}T}]}{Z}\\ \end{aligned}\]

So the thermodynamic stability of the native ground state is quantified as:

\[\begin{aligned} \Delta G &= -K_{B}T\ln(\frac{P_N}{P_D})\\ &= E_N + K_{B}T\ln\left[\sum_{E>E_N} n(E) \exp(\frac{-E}{K_B T}) \right] \end{aligned}\]

NOTE: Still on writing.

---

Cited as:

@online{zhu2022funneled,
        title={Conformation 101 - Funneled energy landscape},
        author={Zefeng Zhu},
        year={2022},
        month={May},
        url={https://naturegeorge.github.io/blog/2022/05/funneled-energy-landscape/},
}