Stretching lattice models of protein folding

Abstract

A new class of experiments that probe folding of individual protein domains uses mechanical stretching to cause the transition. We show how stretching forces can be incorporated in lattice models of folding. For fast folding proteins, the analysis suggests a complex relation between the force dependence and the reaction coordinate for folding.

Figure 1

Unfolding and folding times versus end-to-end stretching force. The simulation temperature was 1.4, which is slightly below the folding temperature for this sequence (T_f = 1.51). The unfolding time was defined as the time (in Monte Carlo steps) it took to reach a state of Q <6. We see three different regimes. At very low and very high force the unfolding time is roughly independent of the force. At intermediate forces the relation between the unfolding time and the force appears linear. Also shown is the equilibrium folding time, which is calculated from the unfolding time and the equilibrium constant (τ_f = K_uτ_u). Once the force becomes large enough to destabilize the compact state the folding time grows rapidly.

Figure 2

Stability vs. stretching force. (Upper) The probability of being in the native state P_nat as a function of the stretching force. We can define an equilibrium unfolding force, the force at which P_nat = 0.5, which is approximately 2.7. (Lower) The free-energy difference (ΔG₀) as a function of force.

Figure 3

The unfolding time versus stability plot (Upper) and its negative slope (Lower) vs. K_unfold. The extra point (■) was determined by using a denaturant simulation without any force (17). α corresponds with the transition state ensemble location, α = Q†/Q_native.

Figure 4

Free energy as a function of Q for various pulling force strengths. For weak forces the transition state remains fixed, but for moderately high forces the transition state moves closer to the native state, changing from an initial value of ≈16 to ≈22 by the time the force reaches 4.

Figure 5

Stability and end-to-end distance as a function of pulling force. One interesting point, the system becomes more stable as the force increases initially. This effect is not universal, but is the result of the relation of the end-to-end distance of the native state (l_nat) and the average end-to-end distance of collapsed molten globule states (l_MG). Note the protein unfolds before a substantial increase of the end-to-end distance occurs, indicating that there are possibly two overlapping transitions: an unfolding event and then an elongation event.

Figure 6

A comparison between the simulated results (see Fig. 2) and the theoretical model discussed in the text (see Eq. 3). The energy for the folded state (E_nat = −84) is the exact value for the lattice model and F_mg = −82.2 is chosen to give the appropriate folding temperature for the system in the absence of forces. A simple function is used to represent l_d(N) = (1 + l_mg)(N/N₀)^1/3 + N₀ − 1 − N. A exact value is used for l_nat = 2.7 and l_mg is fitted to a value 2, which is very close to the actual value.

Figure 7

Free energy as a function of the end-to-end separation (l_end) for several different values of the pulling forces. The curves were calculated by using the Monte Carlo histogram technique. Note, the ruggedness at a larger separation is the result of sampling error.

Figure 8

Effective transition state displacement (δl) versus force. The plot is computed from the transition state free energy barrier by: δl = −∂ ΔG^‡/∂F. The barrier height was estimated by using the data in Fig. 1 assuming a simple Kramer’s form for the unfolding time, ΔG^‡ = Tlogτ_u + C. The slight negative values at small force are the result of the fact that this particular native state is stabilized by a small stretching force. The dotted line is a polynomial fit to the data as an aid to the reader.