Associative Pattern Recognition Through Macro-molecular Self-Assembly

Abstract

We show that macro-molecular self-assembly can recognize and classify high-dimensional patterns in the concentrations of N distinct molecular species. Similar to associative neural networks, the recognition here leverages dynamical attractors to recognize and reconstruct partially corrupted patterns. Traditional parameters of pattern recognition theory, such as sparsity, fidelity, and capacity are related to physical parameters, such as nucleation barriers, interaction range, and non-equilibrium assembly forces. Notably, we find that self-assembly bears greater similarity to continuous attractor neural networks, such as place cell networks that store spatial memories, rather than discrete memory networks. This relationship suggests that features and trade-offs seen here are not tied to details of self-assembly or neural network models but are instead intrinsic to associative pattern recognition carried out through short-ranged interactions.

Appendices

Appendix 1: Blurring Mask

We use a blurring mask to easily visualize how closely a concentration pattern $\vec {c}_a$ matches a memory $m_\alpha$. At each pixel (i.e., position) x of a 2-d memory $m_\alpha$, we consider all species in a square box of area q centered at that pixel and find the overlap $\chi (x)$ by finding the fraction of species with high concentration. At position x, we then display a gray scale value $(1-\chi (x))*g_{random} + \chi (x)*g_{correct}$, where $g_{correct}$ is the correct gray scale value and $g_{random}$ is a randomly chosen grayscale value. That is, if every species around x has high concentrations, then $\chi (x) = 1$ and we display the correct pixel. If all species are of low concentrations, then $\chi (x) = 0$ and we display a random gray scale value. Thus, the image displayed is a measure of the contiguity of high concentration species in pattern $\vec {c}_a$ in the spatial arrangement corresponding to memory $m_\alpha$.

In the paper, we use the $\chi$ to compute the rate of nucleation with that region. So q must be larger than the critical nucleation seed at concentrations $c_{low}$ and $c_{high}$. But q must also be small enough, so the $\chi (x)$ computed is representative of the whole region of size q. For the images presented here, $q = 8$ was found to satisfy all of these conditions.

With these conditions, the blurring mask provides an immediate visual predictor of nucleation since the blurriness determines the nucleation rate through an exponential function (see Eq. 4). If a memory appears coherent over a region, the spontaneous nucleation rate is high in that region and that memory will be rapidly self-assembled. Blurry images have greatly suppressed nucleation rates.

Appendix 2: Gillespie Simulation for Self-Assembly

In our model, particles have distinct binding sites with specific interactions. In the 1D model, each particle has two distinct binding sites, left and right, while in the 2D model, each particle has four distinct sites. In our simulations, particles cannot “turn around”; i.e., incoming particles in Fig. 7 always present their left binding site.

We use a Gillespie algorithm to simulate the kinetics shown in Fig. 7 and Fig. 3; the Gillespie algorithm provides an exact way of measuring physical time using discrete time simulations. At each time step, we consider two kinds of processes: (1) all possible additions of a molecule of any species i to the boundary of the growing structure (2) all possible removals of molecules at boundary of the growing structure

We compute the free energy difference associated with each of these outcomes. For example, if species i is added to the boundary of the structure, the free energy changes by

$$\begin{aligned} \Delta F = - n E_0 + \mu _{i} \end{aligned}$$

(12)

where $\mu _i$ is the chemical potential of species i and n is the number of bonds made by species i with its neighbors in the structure (which is, in turn, determined by the interaction matrix $J^{tot}_{ij}$ described in the main text).

We assume that the kinetic rate associated with such a process is $k = e^{-\frac{\Delta F}{2}}$. Removal rates are calculated with the same formula but with the appropriate sign reversals. One of these reactions is randomly chosen and implemented with probability proportional to the corresponding kinetic rate. Physical time is incremented by $t \rightarrow t + \frac{1}{\sum _a k^+_a + \sum _x k^-_x}$, in accordance with Gillespie’s prescription.

The real time associated with the final frame of simulations in Fig. 3 are $\vec {c_1}$: 221221; $\vec {c_2}$: 168449; $\vec {c_3}$: 141202; $\vec {c_4}$: 355121; $\vec {c_5}$: 182573; $\vec {c_6}$: 389395 (all times in arbitrary units). These times are measures of nucleation time in one simulation run from which Fig. 3 was made; growth is much faster than nucleation in our parameter regime.

In this Gillespie simulation, we forbid nucleation at multiple points in the box. Instead we seed the initial structure by one random tile appearing and do not let it disappear. Such a choice does not affect the results and is necessary since our simulations do not include diffusion. Seeding with a single monomer amounts to following the fate of a generic monomer in the solution. Further, we repeat the simulation and find that the results for a given initial overlap stays the same.

Finally, Eq. 12 can be written in a more intuitive form by defining $f \equiv q E_0 - \mu > 0$, the “force” driving self-assembly forward (where q is the range of interactions in the 1-d model). If a species binds with energy $E_0$ to only $n = qw$ of the q molecules in the growing tip in Fig. 7, then Eq. 12 reduces to,

$$\begin{aligned} \Delta F = -f + w E_0 \end{aligned}$$

(13)

Thus at low driving forces $f \approx 0$, growth happens near equilibrium since even a molecule making the maximal number q of strong bonds (and thus with $w=0$) binds almost reversibly. At high force $f>0$ (high concentrations), even species that fail to bind strongly with $w > 0$ particles in the tip can bind. We connect Eq. 13 with corresponding equations for neural networks below.

Appendix 3: Neural Network Simulation

We perform kinetic Monte Carlo simulations on place cell networks using paired spin flips; at each discrete time step, we randomly pick an ‘on’ neuron b and an ‘off’ neuron a and attempt to flip their states. The free energy change associated with such a move is,

$$\begin{aligned} \Delta F = -\sum _{c \in clump, c\ne a,b} (J_{ac} - J_{cb}) \end{aligned}$$

(14)

We accept such a move with probability $e^{-\Delta F}$.

In our simulations, we drive the clump [20, 37] by modifying $J_{ij} \rightarrow (1-f/q)J_{ij} + f/q A_{ij}$ by an anti-symmetric component $A_{ij} = - A_{ji}, abs(A_{ij}) = abs(J_{ij})$; this applies a driving force on the clump from left to right in each environment in Fig. 6.

We can build some intuition for the driving forces f and connect with self-assembly by re-writing $\Delta F$ for some low energy moves. The left-most neuron in a clump (e.g., neuron 4 in Fig. 7) is the easiest to turn off, with a cost $J_{off} = q (E_0 - f/q) = q E_0 - f$. Turning on a neuron a to the right of the clump gives an energy $J_{on} = (q - w)E_0$ where $q-w$ is the number of strong right-ward connections with neurons in the clump made by neuron a. Hence the free energy change in moving the clump by turning off the left-most neuron (4 in Fig. 7) and turning on neuron a can be written as,

$$\begin{aligned} \Delta F_{Clump} = -f + w E_0 \end{aligned}$$

(15)

Such a move is most favorable if $w=0$, i.e., if the new neuron a is fully connected to the clump neurons and thus if the clump moves without breaking up (e.g., neuron 8 in Fig. 7).

Comparing Eqs. 15 and 13, we see that low energy transformations of the clump are in correspondence with transformations of the growing structure.