Maziar
Heidari‡
ab,
Vahid
Satarifard‡
ac,
Sander J.
Tans
d,
Mohammad Reza
Ejtehadi
e,
Samaneh
Mashaghi
f and
Alireza
Mashaghi
*a
aLeiden Academic Centre for Drug Research, Faculty of Mathematics and Natural Sciences, Leiden University, Leiden, The Netherlands
bMax Planck Institute for Polymer Research, Mainz, Germany
cMax Planck Institute of Colloids and Interfaces, Potsdam, Germany
dFOM Institute AMOLF, Amsterdam, The Netherlands
eDepartment of Physics, Sharif University of Technology, Tehran, Iran
fSchool of Engineering and Applied Sciences and Department of Physics, Harvard University, Cambridge, Massachusetts, USA. E-mail: a.mashaghi.tabari@lacdr.leidenuniv.nl
First published on 5th July 2017
Linear chains with intra-chain contacts can adopt different topologies and allow transitions between them, but it remains unclear how this process can be controlled. This question is important to systems ranging from proteins to chromosomes, which can adopt different conformations that are key to their function and toxicity. Here, we investigate how the topological dynamics of a simple linear chain is affected by interactions with a binding partner, using Monte Carlo and Molecular Dynamics simulations. We show that two point contacts with a binding partner are sufficient to accelerate or decelerate the formation of particular topologies within linear chains. Computed ‘‘folding-time landscapes” that detail the folding time within the topology space show that such contacts deform these landscapes and hence alter the occupation probability of topological states. The results provide a mechanism by which chain topologies can be controlled externally, which opens up the possibility of regulating topological dynamics and the formation of more complex topologies. The findings may have important implications for understanding the mechanism of chaperone action as well as genome architecture and evolution.
Here, we investigate how a simple interacting molecule (chaperone) guides folding of client polymer chains toward certain topological states and away from others, and how it modulates the kinetics of folding. We design three simple modeling experiments: (I) an interacting molecule perturbs one of the native intra-chain contacts of the client during folding, by increasing or decreasing the binding affinity of the contact sites (see Fig. 1b); (II) an interacting molecule closes a random loop on its folding substrate which does not exist in the native state (see Fig. 1c); (III) an interacting molecule interferes with client folding by end-to-end cyclization of a linear client polymer chain or by closing a half loop on a partially folded substrate (see Fig. 1d). To this end, we perform Kinetic Monte Carlo (KMC) simulations of a linear Gaussian chain to analyze the folding kinetics and use Molecular Dynamic (MD) simulations for population analysis of topological arrangements in the presence and absence of an interacting molecule. Although the excluded volume interaction is not present in the KMC simulations which makes its outcome difficult to compare with the steric-mediated folding processes of a real biopolymer, such a simple model is able to provide insights into the configurational entropy, topological occupancy, and kinetic pathways of the constrained chain relative to a free chain.15,16 Nonetheless, through integrating these complementary approaches, we endeavor to contextualize the role of an external interacting molecule in modulating the circuit topology of the folding chain and regulating its folding kinetics.
(1) |
kd[{i,j}] = νexp(βεij) | (2) |
To describe the KMC procedure for a polymer whose native state has N native contacts, we consider a polymer chain in a state with m contacts {C1,…,Cm}. To move in the conformer space, any of the N − m contacts could form or each of the m contacts could break; the rate constant of each event is calculated based on eqn (1) and (2). In this step, the total rate constant is defined by
(3) |
At each stage of folding in KMC, the association (ka) and dissociation rates (kd) of possible new or old contacts are computed. The corresponding probabilities of each move in the conformer space are given by Pi = ki/ktotal, where Pi denotes the probability of the ith possible move in the conformer space. The time when the next event happens is given by tevent = −ln(ξ)/ktotal, where ξ is a random variable with values in [0,1]. We start our KMC simulations from a denatured state of the polymer chain (i.e., initially there is no contact). The chain moves in the conformer space and the simulation ends once the polymer finds its native state. The total times of all the events are summed and called the folding time (Tfold). Since the average value of −〈ln(ξ)〉 is equal to unity, we calculate the time of the next event by tevent = 1.0/ktotal. This protocol generates a random walk through the conformer space and allows one to calculate the folding time of a polymer with a known native state.
Using the procedure explained, we simulate a short and long polymer consisting of 50 and 200 monomers under four sets of conditions:
(i) For the two-contact and four-contact experiments, four and eight monomers are randomly chosen from the polymer respectively and they are mutually connected so as to construct the conformer space. For each experiment, the native state is initially fixed by a set of the defined native contacts. The chain is in its native state and fully folded when all native contacts are formed. The permutation in the connectivity is done using the Heap algorithm17 to ensure all conformations are surveyed and this is done for all 5 × 106 conformations with different topological arrangements. This is verified through obtaining equal values in each topological state. The diffusion constants (D) for all the binding sites are identical and these sets of simulations are named control sets.
(ii) The same number of binding sites as the control sets is selected but the diffusion constant of one of the contacts (which is manipulated by the chaperone and shown in Fig. 1b) is set three times less (i.e. 0.3D) or more (i.e. 3D) than the other contacts; these sets of simulations are named (ii-1) and (ii-2), respectively.
(iii) The number of binding sites (contacts), with respect to case (i), increases by two (one) and then the new binding sites have a diffusion constant three times less (0.3D) or more (3D) than that of the other binding sites (D); these sets are named (iii-1) and (iii-2) respectively. We assume that this extra contact is formed by an external molecule which does not exist in the native state and create a random transient loop along the chain. Sets (ii) and (iii) are designed to investigate the effect of the external molecule on changing the folding times of the control set. In case (iii), the folding time of the external contact is not taken into account when it folds lastly.
(iv) Similar to case (i), additional binding sites close a loop on the polymer chain by attaching the terminal residues of the chain (full loop, (iv-1)) or binding one monomer in the middle of the chain and one terminal monomer (half loop, (iv-2)). In these cases, the external contact is not considered in the calculation of topological fractions. The proposed scenario (iii) gives us a comprehensive view towards the kinetic effect of transient loop formation by interacting an external molecule on the folding substrate.
We choose a high unfolding energy barrier in all sets of simulations such that no contact would be able to unfold during the simulations. In all cases, at the beginning of the simulation, the chaperone-induced contact is also introduced in the conformational space of the chain and during the folding process, once it is formed, it is not dissociated during the rest of the simulations. The fraction of three topological relations and their corresponding folding times are calculated. We use a ternary phase diagram to map the topological states and the corresponding folding rates of the polymer chain in the topological fraction space (SPX space). In what follows, we refer to this map simply as the folding time landscape. Due to the scattered points on the ternary plot, we use interpolation techniques to provide a smooth folding time landscape. The corresponding averaged contact orders of each topological set are also calculated. The contact order of two loops with the topology of k is calculated by , where Nk is the number of double loops which are categorized in the topological state k, ΔL(1)i and ΔL(2)i are the monomer separation of each loop and L is the total polymer length.
We next studied how chaperones can affect folding rates by accelerating or decelerating the formation of a native contact or by introducing a transient non-native contact with fast or slow kinetics (compared to native contacts in our control). For doing so, we altered the diffusion constants of binding sites associated with a native contact and asked how that affects the folding time. In the case of decreasing the diffusion constant of a native contact by 0.3D (i.e. case ii-1) (Fig. 3a), as expected, the folding times of all topological states rise. However, when a non-native contact with diffusion constant, 0.3D, is introduced in the contact set of the polymer (i.e. case iii-1) (Fig. 3c), the folding time of the chain decreases with respect to the control set; despite the fact that a non-native contact is formed. Interestingly, although the non-native contact has slower kinetics than the other contacts, overall it expedites the folding process of all the topological states by introducing interfering foldimers in the folding pathway. As is shown in Fig. 3b and d, such a catalyst mechanism reduces the folding time even further when the diffusion constants of the native (ii-1) and non-native (iii-1) contacts are increased three times. Hence, irrespective of whether the interfering contact has slower or faster dynamics with respect to the other contacts, it would enhance the kinetics of the folding process.
Fig. 3 Ternary plots of the rescaled folding time (TfoldD/a2) for a polymer with four contacts in its native state when one native or non-native contact has different diffusion constants with respect to the other polymer contacts. Plots (a)–(d) correspond to cases (ii-1)–(iii-2), respectively. For a better comparison with the control set, the maximum folding time in the colored bars of plots (b)–(d) and the minimum folding time in plot (d) are set to the highest and lowest folding times in Fig. 2a respectively. |
In all preceding analyses, the locations of the non-native binding sites on the chain are random. Thus, the external molecule does not induce any preferential topology on the chain. Subsequently, such inherent randomness in the selection of the binding sites on the chain only changes the overall folding time scales but it does not deform the folding time landscape with respect to the control sets (see Fig. 1e and compare Fig. 2a with Fig. 2 in ESI†) and generally, P-loops are closed slower than X-loops. However, when the external molecule binds both ends of the polymer to form a looped polymer, as discussed in (iv-1), along with global reduction in folding time scales, the folding time landscape deforms such that the maximum peak, which was previously localized near P-loops, moves toward X-loop topologies and it becomes more flattened in the SPX space (Fig. 4a). When a non-native contact connects the middle segment of the chain with one of the chain ends (iv-2) (Fig. 4b), although the global topology of the chain is divided into a half loop and a tail, the loops can randomly form either on the half-loop part or on the tail segment of the chain. Like cases (ii) and (iii), here we observe only a minor change in folding time landscape with respect to the control set; however, the loops in all the topological states fold faster than the loops in the control set (Fig. 4b).
The MD simulations also reveal that in the absence of the interacting molecule and when the chain folds freely in space, for all investigated ranges of contact orders, most of the contacts are arranged in series (Fig. 5). This trend, however, changes when an interacting molecule (chaperone) is present. When the chaperone forces the chain ends to the distance L = 5σ, approximately similar to the full loop case (iv-1), the overall topological fractions change such that the occupancy of series loops is taken over by the parallel loops, due to zipping effects.
Footnotes |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp02145c |
‡ Equal contribution. |
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