Molecular dynamics (MD) and Monte Carlo (MC) simulations are the two most important modeling techniques for the study of biological macromolecules. In MD simulations, the time dependent behavior of atoms and molecules is determined by numerically solving Newton’s laws of motion for a system of interacting particles. Forces between the particles and potential energy are defined by potentials or molecular mechanics force fields. Monte Carlo algorithms, on the other hand, are based on repeated random sampling to get numerical results. In order to obtain the distribution of an unknown probabilistic entity simulations are typically repeated many times.
Metropolis Monte Carlo
We use a Metropolis Monte Carlo algorithm for our simulations. This algorithm leads to thermal equilibrium after a certain number of steps (see the next subsection 3.1.2 for further details) and works as follows. We start from an initial configuration of the particles in a system. The algorithm proceeds by randomly attempting to change the configuration of the particles, i.e. a Monte Carlo move. The move is either accepted or rejected based on the Metropolis acceptance criterion guaranteeing that the sampled configurations are drawn from the Boltzmann distribution with the correct Boltzmann weight. After having either accepted or rejected a move, we compute the quantity in question. The algorithm proceeds by randomly attempting to move about the sample space and eventually, after many moves have been made, it yields a reliable average value of the quantity in question.
Autocorrelation Time
Markov Chain Monte Carlo (MCMC) methods generate a new state based on its previous state. Thus obtained samples by MCMC algorithms are statistically dependent on each other or correlated. The autocorrelation time helps in obtaining statistically independent or uncorrelated conformations in simulations. It is determined using the integrated auto- correlation time τint which is computed using the autocorrelation function C(t) and the normalized autocorrelation function ρ(t). This scheme allows the calculation of the corre- lation between polymer conformations separated by t Monte Carlo steps and described by a particular measure or observable. The autocorrelation function of an observable A(t) is defined as
C(t) = ⟨ A(s + t) · A(s) ⟩s− ⟨ A(s) ⟩2s
and the normalized autocorrelation function is given by ρ(t) = C(t)
C(0), where ⟨ · ⟩sis defined
as the mean of the ensemble at time s.
We use the windowing method by Sokal [64] in order to estimate the integrated auto- correlation time as τint= 1 2 M ∑︂ t=1 ρ(t) .
The integer M is chosen such that M > c · τint. The value c can vary between four for exponential decaying ρ(t) to ten for slower decay [64]. Two subsequently obtained conformations are considered to be uncorrelated when they are separated by more than 5τint steps. In each of our simulations, 10τint steps are prepended in order to equilibrate our artificially generated starting configuration.
We use this windowing scheme of Sokal in favor of simply fitting an exponential model to the autocorrelation function because we are also simulating both large and quite stiff
3.1. Simulation Methods 35 0 1 2 3 4 5 6 7 ·107 0 0.2 0.4 0.6 0.8 1 τc,400 exp = 4.45 · 106 τintc,400= 4.63 · 106 0 0.5 1 1.5 2 2.5 3 3.5 4 ·106 0 0.2 0.4 0.6 0.8 1 τl,250 exp = 4.25 · 105 τintl,250= 4.19 · 105 tMC C (tMC )
Figure 3.1: The main figure shows a rapidly sloping autocorrelation function (in gray) of the mean squared radius of gyration of a linear polymer consisting of N = 250 monomers. The blue curve illustrates an exponential fit and aligns perfectly with the computed values of the autocorrelation function. However, the autocorrelation function of the mean squared radius of gyration of a circular polymer with N = 400 monomers (see inset graph) is not well described by an one-parameter exponential model obeying the functional form C(tMC) = exp(−τexp−1· tMC). The resulting exponential autocorrelation time τexp is illustrated in the graphs and also compared
to the computed integrated autocorrelation time τint.
polymers leading to a slowly decreasing autocorrelation function of the mean squared radius of gyration (see the inset graph in Fig. 3.1).
The Bond Fluctuation Model
The Bond Fluctuation model (BFM) is a well established lattice model for polymers. The BFM includes excluded-volume interactions and preserves the topological state of the polymers by preventing bond crossings. It is a Monte Carlo method characterized by especially high acceptance rates making it a good choice for dense polymer systems. A detailed description of the BFM can be found in [65,66].
A long polymer on a three-dimensional cubic lattice consists of N monomers, numbered from one to N. Each monomer occupies one box (i.e. eight lattice sites) on the lattice and thus the polymer can be described by the set of bond vectors of its comprising monomers {b1, b2, . . . , bN −1}. Volume interactions are integrated into the model by forbidding one
box to be occupied by two or more monomers. As one monomer occupies eight lattice sites, there always has to be at least one empty box between two monomers. The maximum
A
B
0 0.25 0.5 0.75 1 0 0.02 0.04 0.06 0.08 0.1 ϑ [π] P (ϑ ) 2 √5 √6 2.688 3 √10 0 0.1 0.2 0.3 b [l.u.] P (b )Figure 3.2: The a priori probability distribution of A. the bond angles ϑ within the BFM as well as B. those of the bond lengths b. In contrast to the bond lengths, not all possible bond angles are allowed within the BFM fulfilling self-avoidance. The forbidden bond angles violating excluded volume are colored in red whereas allowed ones are depicted in blue.
bond length is restricted to ten, limiting the distance between neighboring monomers and thus preventing the chain from developing gaps. There are further constraints on the bond vectors in order to avoid bond crossings and for ensuring the preservation of the topology of the polymer. On a three-dimensional lattice 108 different bond vectors can be realized. The a priori probability distributions of both the bond angles and the bond lengths are depicted in Fig. 3.2 (forbidden bond angles are colored red). The possibility for fluctuating bonds is a key ingredient of the BFM since this leads to an increased probability for local moves of the monomers resulting in quicker relaxation towards equilibrium.
Dynamic Looping Mechanism
In our simulations we make use of the dynamic loop (DL) model developed by Bohn and Heermann [67]. The DL model is based on the BFM and incorporates the ability of non- adjacent monomers to become linked by a bond vector. Whenever two monomers come close to each other by diffusion, there is a looping probability ploop for them to form an additional bond. When this happens, a crosslink of the fiber is created with a lifetime, drawn from a Poisson distribution with mean value τ. Thus, loops can form and dissolve dynamically. The size of the loops is restricted, monomers must have at least a genomic distance of three to be able to form loops. The maximum allowed size of the loops as well as the number of bonds starting from one monomer can be restricted.
The dynamic and probabilistic crosslinking mimics the effect of surrounding proteins which mediate the process of loop formation. It causes a coiling and local collapsing of the chromatin fiber, which is anticipated to have implications on the shape and the mechanical properties of the polymer.