Bead Spring System: Results

Before the inclusion of the new coupled kinetics, we verified that the dynamical simu- lation works as a standalone simulation by performing a number of standard tests on the bead-spring system. Our two beads, each with an associated drag λ = 5.65kg.s−1

Chapter 4. Kinetic Scheme 97

Figure 4.4: The average potential energy trace of the 1D dumbbell model with no coupled kinetics.

are connected by a spring with spring constant k = 1.8849 × 10−3Nm−1 and equilib- rium length l = 3nm. The choice of spring constant was designed to give a fluctuation time constant of τ = 30ns for testing purposes. We performed an ensemble of 1000 simulations for 100µs to fully capture the dynamics of all available types of motion for this system.

For this system, we firstly checked that each individual simulation equilibrated by measuring the average potential energy of each simulation as an average over the full 100µs, corresponding to 10000 simulation frames. With an expected value hU i = 0.5kBT , the average error for each simulation was (1.37 ± 1.03)%. A single simulation

can be seen equilibrating in Figure4.4, which converged to hU i = 0.5013kBT , a 0.03%

error compared to the expected value.

We also calculated the diffusion of the entire dumbbell with respect to its starting position by calculating hxc− xc,0i at each time t averaging over the entire ensemble

of 1000 simulations. We obtain a diffusion constant D = 74.08nm2/ns, an error of 1.93% compared to the expected value calculated from Equation 4.33. Finally, the variance in the length of the spring was calculated at each time step from the ensemble of simulations. This equilibration of variance is shown in Figure 4.6, where we see that the convergence is almost instantaneous over this time-scale, given the size of our

Chapter 4. Kinetic Scheme 98

Figure 4.5: The diffusion trace of the 1D dumbbell model with no coupled kinetics.

simulation time step. We obtain the average extensional variance using a least-squares fit to a straight line, giving us h(x0 − l)2_{i = 218.89˚A}2_{, a 0.39% error compared with}

the theoretical value.

Now that we have shown that the underlying dynamical simulation functions correctly and within a reasonable error tolerance, we can define multiple kinetic states with associated transition rates and couple them to the dynamical model. We chose, some- what arbitrarily, to define four states with the parameter sets kα and lα as shown in

Table 4.1. These parameters are representative of the biological mesoscale [101] but not specific to any real system. In addition to the states, we must define the mesoscale

State (α) k(pN/nm) l(nm)

1 1 5

2 2 3

3 3 2

4 4 1

Table 4.1: The parameters defining each one of the kinetics states within the bead- spring system.

average rates, Rαβ, at which the states transition between one another. As we saw

from Equation 4.6, the detailed balance condition means that occupation properties are related to the relative rate proportions only, and not their individual magnitudes.

Chapter 4. Kinetic Scheme 99

Figure 4.6: The variance in length between beads for the 1D dumbbell model.

We chose four occupation probabilities in advance, 0.1, 0.2, 0.3 and 0.4, and used the detailed balance conditions between each pair of states to define a set of rates that should correspond to the pre-defined distribution of mesostate occupation. The two sets of values shown in Table 4.2have this property, with the faster set simply being a multiple of the smaller set, thus preserving the relative rate conditions required for detailed balance whilst at the same time having a faster equilibration time-scale. These rates have been largely accelerated when compared with experimental rates (chemi- cal reaction rates, for example) in order to show the convergence of the occupation probabilities within a reasonable amount of simulation runtime.

Set 1 Set From / To 1 2 3 4 From / To 1 2 3 4 1 N/A 30 20 10 1 N/A 300 200 100 2 40 N/A 20 10 2 400 N/A 200 100 3 40 30 N/A 10 3 400 300 N/A 100 4 40 30 20 N/A 4 400 300 200 N/A

Table 4.2: The two sets of average transition rates Rαβ between kinetics states

within the bead-spring system, with units of MHz.

Figure 4.7 shows the convergence of occupation probabilities for the two simulations resulting from each of the sets of kinetics rates. These initial simulations ran without the mechanical energy modifications to the transition rates shown in Equation 4.42

Chapter 4. Kinetic Scheme 100

(a) Slow set of kinetic rates (b) Fast set of kinetic rates Figure 4.7: Running averages of the kinetic state occupation probabilities emerging from the two different sets of transition rates. These rates were constant throughout

the simulation, i.e. unmodified by the dynamic mechanical energies.

i.e. r(~x)αβ = r0αβ = Rαβ. In each, we see that the same occupation probabilities are

reached but at different rates, consistent with the differing magnitudes of the two sets of kinetic rates. Additionally, these converged probabilities are consistent with the analytical solution of each of the detailed balance conditions between states.

Figure 4.8shows similar convergence properties for a simulation using the slow set of kinetic rates, but this time including the energy modification term in Equation 4.42, therefore utilising the full kinetic framework we have developed. We see that these continuously varying kinetic rates seem to have converged to the global averages faster compared to Figure 4.7a. However, this is simply an artefact of the single simulation and not a general property of including energy modifications to the kinetic rates. The important outcome is that the set of base rates r_αβ0 were not equal to the average rates throughout the simulation, yet the exploration of phase space and associated energy modifications that form the varying rates r(~x, ~p)αβ gave us exactly the occupation

probabilities we expected, validating the technique. The set of r0_αβ values calculated using Equation4.42 are shown in Table4.3.

Set 1 From / To 1 2 3 4 1 N/A 43.00 42.44 34.28 2 57.34 N/A 23.94 16.86 3 84.88 35.91 N/A 11.93 4 137.12 50.58 23.86 N/A

Table 4.3: The set of (slow) base transition rates r0αβbetween kinetic states within

Chapter 4. Kinetic Scheme 101

Figure 4.8: Running averages of the kinetic state occupation probabilities emerging from the slow set of transition rates. These rates were modified by the dynamic energy

changes throughout the simulation.