Long-lived metastable knots in polyampholyte chains

Abstract

Knots in proteins and DNA are known to have significant effect on their equilibrium and dynamic properties as well as on their function. While knot dynamics and thermodynamics in electrically neutral and uniformly charged polymer chains are relatively well understood, proteins are generally polyampholytes, with varied charge distributions along their backbones. Here we use simulations of knotted polymer chains to show that variation in the charge distribution on a polyampholyte chain with zero net charge leads to significant variation in the resulting knot dynamics, with some charge distributions resulting in long-lived metastable knots that escape the (open-ended) chain on a timescale that is much longer than that for knots in electrically neutral chains. The knot dynamics in such systems can be described, quantitatively, using a simple one-dimensional model where the knot undergoes biased Brownian motion along a "reaction coordinate", equal to the knot size, in the presence of a potential of mean force. In this picture, long-lived knots result from charge sequences that create large electrostatic barriers to knot escape. This model allows us to predict knot lifetimes even when those times are not directly accessible by simulations.

Fig 1. Charge sequences studied here.

Left: “randomly” charged chains, with sequences labeled q1 to q6. Right: chains with diblock charge distributions, with same-charge block lengths n ranging from 1 to 25. Blue and yellow color stripes represent positively and negatively charged segments.

Fig 2. Simulated time dependence of the knot size (defined in the methods section) for an electrically neutral polymer chain, chains with diblock charge distributions (with block lengths n = 1 and 25) and one of the randomly charged polyampholyte chains (q4 sequence from Fig 1).

The knots in the polyampholyte chain and in the diblock copolymer with n = 25 remain intact over the entire simulation time, whereas the knots in the neutral chain and in the chain with alternating positively and negatively charged monomers (n = 1) grow in size or diffuse along the chain towards the ends until they untie. Inset: time evolution of the knot boundaries. Here n_f and n_l are the first and the last monomers participating in the knot, where the total chain length is N = 500. Data are shown for the neutral chain and for the chain with alternating charges. The unit of time τ is defined in Section 2.

Fig 3. Relaxation times of the end-to-end distance of PA chains plotted against knot lifetimes for n = 1, 5, 10 and 15 diblock (blue circles) and for the q1, q2 and q3 charge sequences (red circles).

Fig 4. Knot in a polymer chain.

Monomers in red represent the knot region. a) The knot size is defined as the number of monomers n_f -n_l in the knot region, and the knot position is defined as (n_f +n_l)/2, where n_f and n_l are the first and the last monomers participating in the knot, with the monomers being numbered sequentially from one chain end (n = 1) to the other (n = N). b) A snapshot of a typical knotted chain configuration (here the sequence is q1 after simulation begins. PA chains tend to collapse to dense structures as a result of electrostatic interactions.

Fig 5. The product of the mean first passage time (MPFT) and the diffusion coefficient D estimated using Eq (1) and plotted against the mean knot lifetime obtained from simulations.

The slope of the linear fit (dashed blue line) gives an estimate for the diffusion coefficient, D = 0.6 σ² / τ, for the reaction coordinate x equal to the knot size.

Fig 6. Potential of mean force as a function of the knot size for a polymer chain with a diblock charge distribution (n = 20).

As x → 0 this potential, of course, must diverge preventing the knot from shrinking to zero size, but this high-energy region is not sampled by the simulation. Thus, for the purpose of evaluating the mean first passage time using Eq 1, the computed potential was extrapolated toward smaller values of x (i.e. tighter knots) such that it diverges for $x\to 0$ (green dashed line). The initial knot size is x_A; the knot is considered untied when the coordinate x reaches a value x_C−both of these values are indicated as vertical red dashed lines.