Statistical physics of the symmetric group

Ordered chains (such as chains of amino acids) are ubiquitous in biological cells, and these chains perform specific functions contingent on the sequence of their components. Using the existence and general properties of such sequences as a theoretical motivation, we study the statistical physics of systems whose state space is defined by the possible permutations of an ordered list, i.e., the symmetric group, and whose energy is a function of how certain permutations deviate from some chosen correct ordering. Such a nonfactorizable state space is quite different from the state spaces typically considered in statistical physics systems and consequently has novel behavior in systems with interacting and even noninteracting Hamiltonians. Various parameter choices of a mean-field model reveal the system to contain five different physical regimes defined by two transition temperatures, a triple point, and a quadruple point. Finally, we conclude by discussing how the general analysis can be extended to state spaces with more complex combinatorial properties and to other standard questions of statistical mechanics models.

Figure 1

Folding vs design (or inverse folding) problems. The protein folding problem is concerned with determining the three-dimensional structure produced by a particular sequence of amino acids. The protein design problem (which motivates the current work) is concerned with finding the sequence(s) of amino acids, which yields a given three-dimensional polypeptide structure. A number of approaches to the design problem are given in [1].

Figure 2

Free energy for “noninteracting model.” For ${\lambda }_0>0$, the Landau free energy of the system as a function of $j$, the number of incorrect components in $\vec{\theta }$, is always convex with a single global minimum. Because $j\ge 0$, the $j<0$ domain of each plot (dashed section) is inaccessible. For sufficiently low temperatures, the minimum is at $j=0$, but as we increase the temperature beyond Eq. (18), the free energy curve moves to the right (but retains its functional form) and the new minimum is at a $j>0$ value. Actual plots of $F\left(j\right)$ for $j<0$ require us to replace the combinatorial term $\left(\genfrac{}{}{0pt}{}{N}{j}\right)d_j$ with its corresponding $\mathrm{\Gamma }$ function expression.

Figure 3

Possible functional forms of Eq. (30). We note that the stabilities that define the $j=0$ and $j=N$ points are not thermodynamic stabilities [namely, they do not arise from the $f^{\prime }\left(j\right)=0$ condition]. Rather, since the spectrum of $j$ values is bounded below by 0 and above by $N$, owing to these boundary conditions the system can become trapped in ordinarily unstable parts of the free energy curve. The colors match the color of the associated region of parameter space in Fig. 4.

Figure 4

${\lambda }_1-{\lambda }_2$ parameter space for interacting mean-field system. We set $\beta =1$ and $N=100$. We followed the Monte Carlo procedure outlined in the notes for $10000$ points. In the figure we also denoted the analytic lines [i.e., Eq. (C3), Eq. (C7), and Eq. (C10)] which define the separation between the phases. The colors correspond to the colors of the free energy curve in Fig. 3.