Assessing the Polymer Coil-Globule State from the Very First Spectral Modes

The determination of the coil-globule transition of a polymer is generally based on the reconstruction of scaling laws, implying the need for samples from a rather wide range of different polymer lengths $N$. The spectral point of view developed in this work allows for a very parsimonious description of all the aspects of the finite-size coil-globule transition on the basis of the first two Rouse (cosine) modes only, shedding new light on polymer theory and reintroducing well-established spectral methods that have been surprisingly neglected in this field so far. Capturing the relevant configuration path features, the proposed approach enables one to determine the state of a polymer without the need of any information about the polymer length or interaction strength. Importantly, we propose an experimental implementation of our analysis that can be easily performed with modern fluorescent imaging techniques and would allow differentiation of coil or globule conformations by simply recording the positions of at least three discernible loci on the polymer.

Figure 1

(a) On lattice MC simulation snapshots for $N=431$ and $ϵ=0$ (orange), $ϵ=0.36$ (black), and $ϵ=0.8$ (blue). (b) Projections $r_x(n)$ along the spatial $x$ axis of a set of 20 configurations, $n$ being the monomer index. Colors refer to the same parameters as in (a). Dotted lines correspond to the $\pm 2.5{\sigma }_x(n)$ envelopes. (c) Logarithm of the square discrete cosine transform of the 20 trajectories of (b), $\ln |X_p{|}^2$, as functions of the logarithm of the mode number $\ln p$ (thin lines) and their average calculated over the ensemble of 16 384 simulated configurations and over the three spatial directions (thick lines and dots). Standard deviations are smaller than the symbol size. Groups of curves are vertically shifted for clarity. Dashed lines correspond to slopes $-2.2\equiv -(1+2\nu )$, $-1.3$, and 0 (from top to bottom). (d) Averaged power spectral density (PSD) obtained by applying the same procedure as in (c) to equivalent simulations with $ϵ$ going from 0 to 0.6 (see color bar). Dashed lines correspond to slopes $-(1+2\nu )$ (top) and 0 (bottom). Inset: the slope ${\alpha }_N(ϵ)$ of the log-log representation of the PSD, estimated from the first two modes, as a function of $ϵ$.

Figure 2

Top panel: log-log plot slope ${\alpha }_N(ϵ)$ estimated on the first two modes of power spectral density (dots) for all polymers sizes $N$ from 101 to 3000 (right to left). For each $N$, ${\alpha }_N(ϵ)$ is fitted by a sigmoid $S(x)=D+A(x-B)/\sqrt{1+C(x-B{)}^2}$ (lines). Colored circles are inflection points, defining ${ϵ}^I(N)$. Blue diamonds indicate the ${\alpha }_N(ϵ)=-2$ condition, defining ${ϵ}_{\theta }^{\mathrm{RW}}(N)$. Inset: ${\alpha }_N(ϵ)$ relative standard deviation, $\mathrm{\Delta }{\alpha }_N(ϵ)/{\alpha }_N(ϵ)={\sigma }_{X_1^2}/⟨X_1^2⟩+{\sigma }_{X_2^2}/⟨X_2^2⟩$. Triangles indicate the position of maxima ${ϵ}^F(N)$, determined by fitting the top of the pics by a Gaussian function. Bottom panel: corresponding phase diagram in the $(ϵ,N)$ phase space. The orange to blue color scale correspond to values of ${\alpha }_N(ϵ)$ from $-2.2$ to 0 (see color bar). Inflection points (colored circles) and fluctuation maxima (colored triangles) are shown and fitted by curves of expression ${ϵ}_{\theta }(N)=a_1{N}^{-1/2}+a_2(2N{)}^{-1}+{ϵ}_{\theta }$ (green and white lines, respectively). The red dashed line is the critical curve of the same expression, with the parameters obtained by Vogel, Bachmann, and Janke [13]. Fitting parameters are given in Table S1 [18]. Both panels: Samples from the sectors $(N,ϵ)\in [700;3000]\times [0.55;1.0]$ and $[300;700]\times [0.7;1.0]$ did not meet statistical relevance and were omitted from the results.

Figure 3

(a) Power spectral densities obtained by successive decimation of a $N=431$ chain for $ϵ=0.0$ (orange, coils), $ϵ=0.32$ (black), and $ϵ=0.46$ (blue, globules). Decimation goes from $M=N$ to $M=3$ (longer to shorter curves). (b) The asymptotic and critical values of ${\alpha }_{N,M}$ obtained from the sigmoid fit of the decimated spectra, as a function of $M$ (dots and lines): ${\alpha }_C(M)={\alpha }_{N,M}(0)$ for coils, in orange; ${\alpha }_I(M)={\alpha }_{N,M}({ϵ}^I)$ for inflection points, in black; ${\alpha }_G(M)={\alpha }_{N,M}(ϵ\to \infty )$ for globules, in blue. These values are averaged over $N$. Shaded regions correspond to $2{\sigma }_{\alpha }$ confidence intervals for different sample sizes from 256 (larger intervals) to 2048, calculated by propagating the statistical variance through Eq. (5). We used a dataset of 16 384 configurations for each $(N,ϵ,M)$ condition.