Free energy and protein folding

Grouping microstates into states¶

The word state is overloaded and is easilyt one of the more slippery defined concepts in physics/chemistry and biology. SO let us try to be more precise:

A microstate $i$ : a single frozen snapshot of atomic configurations (in CM). For instance instanenotus positions and momenta of all atoms in Ar gas.
A state, $A$ : a collection (fairly large) of microstates. For instance open vs colsed states of a polymer.

Relative populations of states¶

Probabilities are what we observe experimentally. To develop intuition about how energies, entropies, and free energies influence the probabilities of states, let us take a generic 1D model of the energy of a system that has a parabolic shape around some stable states A and B. We will use results obtained in NVT ensemble lecture to derive ratio of probabilities fo macrostates

\frac{p_A}{p_B} = \frac{\int_{V_A} e^{-\beta U(x)}}{\int_{V_B} e^{-\beta U(x)}} = \frac{Z_A}{Z_B} = \frac{e^{-\beta F_A}}{e^{-\beta F_B}}= exp(-\beta \Delta F)

(1)

Z_A = \int_{V_A} e^{-\beta U(x)} \approx \int^{+\infty}_{-\infty} e^{-\beta \big[ U^0_A+ \frac{\kappa_A}{2}(x-x_A)^2\big]} = \Big(\frac{2\pi k_B T}{\kappa_A} \Big)^{1/2} e^{-\beta U^0_A}

(2)

F_A(T) = -k_B T log Z_A = U^0_A -k_B T log \frac{1}{V}\Big(\frac{2\pi k_B T}{\kappa_A} \Big)^{1/2}

(3)

\frac{p_B}{p_A} = \Big(\frac{\kappa_A}{\kappa_B} \Big)^{1/2} e^{-\beta (U^0_B-U^0_A)}

(4)

import numpy as np
import scipy
import matplotlib.pyplot as plt

def E(x, k, x0=0, U0=1):
    return U0 + k*(x-x0)**2

def FE(k, kbT=1, U0=1):
    return U0 - kbT * np.log(2*np.pi*kbT/k)**0.5

x = np.linspace(-100, 100, 1000)
Ea = E(x, k=1,   x0= 0, U0=0.)
Eb = E(x, k=10,  x0= 0, U0=-4.)

kbT = np.linspace(0.1, 10, 1000)
Fa = FE(k=1,  U0=0,  kbT=kbT)
Fb = FE(k=10, U0=-4, kbT=kbT)

#fig, ax  = plt.subplots(ncols=2)
# Plot energies

# Plot free energies

/tmp/ipykernel_3244/664538776.py:9: RuntimeWarning: invalid value encountered in sqrt
  return U0 - kbT * np.log(2*np.pi*kbT/k)**0.5

Protein folding problem with 1D potential¶

As crazy as it sounds we can learn something about protein folding by using a simple 1D potential:

def U_fold1D(ratio=1/25, deltaU=50, T=0.1):
    
    '''1D energy function, where
    rationL: fraction of folded vs unfolded states (must be less than 1)
    deltaU: energy difference between unfolded and folded states (must be positive) in units of kb = 1
    T: temperature (must be possitive)
    OUTPUT: x coordinate, U(x) potential, pf, pu" fractions of folded and funfolded states respectively. 
    '''
    assert np.any(T) > 0 
    assert ratio < 1
    assert deltaU > 0
    
    x = np.linspace(0,1,1000)
    U = np.linspace(0,1,1000)
    
    U[x < ratio] = 0
    U[x > ratio] = deltaU
    
    pf = ratio/(ratio+np.exp(-deltaU/T))
    pu = 1 - pf
    
    U[0] = U[-1] = 100 # set end states to be high
    
    return x, U, pf, pu

x, u, pf, pu = U_fold1D()

fig, ax  = plt.subplots(nrows=2)

ax[0].plot(x,u, lw =3)
ax[0].set_xlabel('$x$')
ax[0].set_ylabel('$U(x)$')

ax[1].plot(x,np.exp(-u), lw=3, color='orange')
ax[1].set_xlabel('$x$')
ax[1].set_ylabel('$p(x)$')

T = np.linspace(0.1,100,1000)
x, u, pf, pu = U_fold1D(T=T)

plt.plot(T,pf,lw=3, color='green')
plt.plot(T,pu,'--',lw=3, color='red')

plt.xlabel('Temperature, T',fontsize=15)

plt.ylabel('Fraction of (un)folded',fontsize=15)

plt.legend(['folded','unfolded'],fontsize=15)