Kinetics of Polyampholyte Dimerization: Influence of Charge Sequences

Abstract

Polyampholytes (PAs) exhibit complex behaviors in various environments influenced by their charge distribution. This study focuses on the kinetics of dimerization of PAs, aiming to elucidate the underlying mechanisms and clarify relevant characteristics of the charge sequence. We focus on PAs with non-zero net charges, employing molecular dynamics simulations and theoretical analyses to examine how charge sequences influence the rates of dimer formation and dissociation. Our findings reveal that the charge sequence of tails and the blockiness of the minority charge group markedly influence the kinetics of dimerization: large blockiness and tails with a high number of majority-type charges slow down the dissociation of dimers. Additionally, the presence of an extended (central) block of the majority charge promotes structural diversity. Within dimer states, blocks alternate between intra- and inter-chain contacts. The duration times in the dimer states are significantly longer than the typical dwell times of block inter-contacts, with a notable extension when multiple blocks are engaged. Intrinsically disordered proteins (IDPs) play crucial roles in cellular functions, primarily due to their ability to undergo rapid conformational changes and form transient complexes. These properties largely depend on the sequence of charged residues. We provide insights into the fundamental principles governing the structural and dynamical properties of polyampholytic IDP, emphasizing the importance of sequence-specific effects on both aggregation and dissociation.

Keywords: IDP; dimerization; polyampholytes; polyelectrolytes.

Figure 1

Fifty PA sequences with Q = 8 are organized into groups based on the blockiness of the minority charge type. Each sequence consists of 100 monomers, with 34 charged units: minority charges are represented in blue, while majority charges are in orange. For clarity, the majority and minority charge types are denoted by + and −, respectively. The lengths of the charge blocks, including all minority blocks and majority-type blocks longer than two units, as well as the corresponding CNC (Central Net Charges) values, are indicated.

Figure 2

Typical dimer conformations (a–c) and a pair of unimer conformations (d–f) obtained from MD simulations for two PA chains with non-vanishing net charges of $Q=8$ are illustrated. Sequences 17 and 29 in (a,b,d,e) contain three blocks of triple minority charges ($(-3)\times 3$), while sequence 46 in (c,f) contains two blocks of quadruple minority charges ($(-4)\times 2$). In these visualizations, red represents majority charges (+) and blue represents minority charges (−).

Figure 3

(a) Three eigenvalues ${\lambda }_1\le {\lambda }_2\le {\lambda }_3$ of inertia tensor for 50 sequences. The eigenvalues are obtained for dimer states and averaged over throughout the simulation times. (b) Distributions of eigenvalues ${\lambda }_1$ and ${\lambda }_3$ for three sequences (seq.13(A1), seq.41(B2), seq.22(C4)) with the CNC values of 5.

Figure 4

Contact maps for representative sequences: (a) seq.46 (A2) and (b) seq.10 (B1). The charge sequence is shown in top rows and most left column, where blue and orange colors indicate minority and majority charge types, respectively. The intensity of the green color represents the contact frequencies, as indicated by the scale bar. The left panels display inter-chain contact maps, while the middle panels show intra-chain contact maps in the dimer states The right panels depict the difference in the intra-chain contact probability between the dimer state and the unimer state.

Figure 5

The survival probability (a) $S_d$ for dimer and (b) $S_u$ for unimer states for some representative sequences. Exponential fits in insets are displayed in different colors for each blockiness group: seq.19 (A1, yellow), seq.16 (B3, purple), seq.4 (B2, blue) and seq.13 (B1, green). For the presentation, the population is reset as 1 at time t = 250$\tau$, that is 35% of the initial population for dimers and 17% for unimers.

Figure 6

(a) The blockiness dependence of the average dwell times $\langle t_d\rangle$ for the dimer state and $\langle t_u\rangle$ for the unimer state considering those that survived longer than $250\tau$. There is anti-correlation between unimer and dimer dwell times for block sequences. (b) The average values of dimer propensity within each group are presented in the right panel for comparison.

Figure 7

(a) The correlation between CNC and the average dwell times is shown for various blockiness groups in (a) and with a specific focus on group A1 in panel (b). Symbols represent different groups: A1 (◦), B3 (△), B2 (◇), B1 (☐), C4 (◁), C3 (▽). The dwell times $\langle t_d\rangle$ at the dimer state are represented as open symbols, while the dwell times $\langle t_u\rangle$ for the unimer state are shown as filled grey symbols. (c) The average dwell times $\langle t_d\rangle$ highlighted for selected subgroups: 4 sequences with $(-4)\times 1,(-2)\times 1$ (red ◦); 3 sequences with $(-3)\times 3,(-2)\times 0$ (navy △); 5 sequences with $(-3)\times 1,(-2)\times 2$ (light blue ☐); 4 sequences with $(-2)\times 4$ (light green ◁).

Figure 8

(a) Schematic illustration showing the reaction coordinate r as the center-to-center distance between unimers. (b) The probability distribution $P\left(r\right)$ of finding another unimer’s center within a spherical shell defined by the range $[r,r+dr]$ is plotted as $-logP\left(r\right)$ with an arbitrary shift for three sequences of Group B3. The positions of the two minima, $r_1$ and $r_2$, represent the stable unimer and dimer states, respectively. (c) Potential profiles $V\left(r\right)$ derived from the probability density.

Figure 9

Comparison of theory (FP calculation) and simulations for sequences indicated in the x-axis ticks. (a) Dimerization times and (b) Dissociation times.

Figure 10

(a) Long time (∼$1.2\times {10}^6\tau$) accumulated contact map and (b) a time trajectory of two-chain states switching between dimer states and unimer states for seq.13. The snapshots display the conformations at the beginning, middle, and end of the dimer state, labeled as S, M, and E, respectively, with the unimer state marked as U. (c) Three dynamic contact maps, accumulated for 250$\tau$ are obtained at the times indicated as S, M, and E. The intensity of the color bars represents contact probability on a scale from 0 to 0.3.

Figure 11

(a) Sequence 39 with two tagged minority charge blocks: a Block of $(-2)$ and a Block of $(-4)$. (b) The intercontact kinetics of the two tagged blocks: Distributions of dwell times $p_1\left(t\right)$ (not normalized) for the tagged block-chain contacts, with the slopes of the fitted lines indicating the kinetics of contact durations: $-1.5$ for Block $(-2)$ and $-1.0$ for Block $(-4)$. Insets display the distributions over longer timescales on a semi-logarithmic scale. (c) The time evolution of the inter-contact points for the tagged blocks is shown for Block $(-2)$ on the left and Block $(-4)$ on the right.