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Proc Natl Acad Sci U S A. 2008 Jun 17;105(24):8238-43.


Casting polymer nets to optimize noisy molecular codes.

Tlusty T.
Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel. tsvi.tlusty@weizmann.ac.il

Life relies on the efficient performance of molecular codes, which relate symbols and meanings via error-prone molecular recognition. We describe how optimizing a code to withstand the impact of molecular recognition noise may be understood from the statistics of a two-dimensional network made of polymers. The noisy code is defined by partitioning the space of symbols into regions according to their meanings. The "polymers" are the boundaries between these regions, and their statistics define the cost and the quality of the noisy code. When the parameters that control the cost-quality balance are varied, the polymer network undergoes a transition, where the number of encoded meanings rises discontinuously. Effects of population dynamics on the evolution of molecular codes are discussed.

Cost of a Code-Table.The cost of a code-table is traditionally measured by the mutual information I between the symbols and the meanings that they encode, I = S ME + S SY − S MS, where S ME and S SY are the entropies of the meaning space and symbol space, respectively, and S MS is the joint entropy of these two spaces. The entropy of meanings S ME is determined by their given distribution P α, S ME = −Σα P α ln P α, and similarly, S SY = −ΣiPi ln Pi = ln(n s). These are constant terms in the cost I that we can neglect, and consider only the joint entropy S MS, which can be optimized by tuning the average partition pattern eij. S MS is simply the entropy of all of the possible coloring patterns as determined by all possible networks and the number of possible ways to color every such pattern. It follows that the cost is therefore minus the coloring entropy I = −S MS = −S C.

Graph Embedding and the Dual Graph.The embedded graph divides the surface into faces or cells, hexagons in our example (Fig. 1). Then, one finds the dual graph by the following correspondence (10): Every vertex in the symbol graph corresponds to a cell in the dual (a triangle in this example) whereas every cell in the symbol graph (a hexagon) corresponds to a vertex of the dual. The correspondence between the edges in the symbol graph and its dual is one-to-one; every edge corresponds to the edge that crosses it in the dual. The resulting dual graph is a triangular lattice. The hexagonal lattice is a regular graph in which all of the vertices have the same coordination number. However, the embedding procedure described here applies to any connected graph whether it is regular or not.

Mean-Field Approximation.The spin probability distribution decouples into a product of independent single-spin distributions, ρ = ∏ijρij(Sij). We use a variational inequality, which sets an upper limit on the spin free energy, F S ≤ F M = 〈ρH S〉 + T〈ρ ln ρ〉, where ρ satisfies probability conservation, 〈ρ〉 = 1. We augment F M with a Lagrange multiplier to account for probability conservation, L = F M + η〈ρ〉, and take the derivative δL/δρij = 0. The resulting distributions are ρij(Sij) = exp(gijSij)/〈exp(gijSij)〉, where the effective fields are gij = ∂H S/∂Sij = ∂(hi + h j)/∂Sij = aibij[∏k(i)≠j(1 + biksik) + ∏k(j)≠i(1 + bjksjk) − 2] with sij = 〈ρSij〉 = 〈Sijρij(Sij)〉. From the n = 0 property, it follows that 〈exp(gijSij)〉 = Σk gijk〈Sk〉/k! = 1 + gij 2/2 and 〈Sijexp(gijSij)〉 = gij. This leads to the self-consistency relations, sij = 〈Sijρij(Sij)〉 = 〈Sij exp(gijSij)〉/〈exp(gijSij)〉 = gij(1 + gij 2/2)−1 (Eq. 7). In a similar fashion, we obtain the mean-field approximation for the free energy, F S ≈ F M = −Σihi + Σi−j gijsij − Σi−j ln(1 + gij 2/2), and it is easy to verify that Eq. 7 defines the extremum ∂F S/∂sij = 0. Eq. 7 is analogous to the self-consistency relation of an Ising magnet, s = sinh(gs)/cosh(gs); the different form is due to the truncation of the power-series expansion of the hyperbolic functions thanks to the n = 0 property. Solving Eq. 7 for gij as a function of sij, we find that the solution can be described graphically as the points where the function eij = gijsij/2 crosses the ellipse 2sij 2 + (2eij − 1)2 = 1 (Fig. 3 A).

Use of Euler's Characteristic.When we apply Eq. 3, an underlying assumption is that the embedding is cellular (10); i.e., every meaning island is homeomorphic to an open disk. This is not necessarily true when the number of islands is less than the number of holes in the surface, that is the genus, γ = 1 − χ/2. In this case, some islands are expected to wrap between two holes and therefore are not homeomorphic to a disk. However, in the “thermodynamic limit” of many islands per hole, n f ≫ |χ|, the embedding is mostly cellular and is well approximated by Eq. 3.