Colloquium: Toward living matter with colloidal particles

A fundamental unsolved problem is to understand the differences between inanimate matter and living matter. Although this question might be framed as philosophical, there are many fundamental and practical reasons to pursue the development of synthetic materials with the properties of living ones. There are three fundamental properties of living materials that we seek to reproduce: The ability to spontaneously assemble complex structures, the ability to self-replicate, and the ability to perform complex and coordinated reactions that enable transformations impossible to realize if a single structure acted alone. The conditions that are required for a synthetic material to have these properties are currently unknown. This Colloquium examines whether these phenomena could emerge by programming interactions between colloidal particles, an approach that bootstraps off of recent advances in DNA nanotechnology and in the mathematics of sphere packings. The argument is made that the essential properties of living matter could emerge from colloidal interactions that are specific—so that each particle can be programmed to bind or not bind to any other particle—and also time dependent—so that the binding strength between two particles could increase or decrease in time at a controlled rate. There is a small regime of interaction parameters that gives rise to colloidal particles with lifelike properties, including self-assembly, self-replication, and metabolism. The parameter range for these phenomena can be identified using a combinatorial search over the set of known sphere packings.

Figure 1

(a) Identical particles with attractive short-ranged isotropic interactions can form many geometrically different, rigid clusters of various sizes $N$. Their number grows rapidly with $N$ (second column of numbers). These clusters are ground states of a system with $N$ particles and are observed in self-assembly experiments. (b) Coating the particles with various DNA strands creates different particle species (top row). The choice of DNA strands determines whether the interactions are attractive or repulsive (bottom three rows). Any cluster geometry can be made the unique ground state in the self-assembly of $N$ particles, if the particle species are appropriately chosen. A matrix represents the chosen set of pairwise interactions between the species (the alphabet), where attractions are represented as gray and repulsions as white (second row). Maximal alphabets (see main text) for the two $N=6$ clusters, the octahedron and the polytetrahedron (third row). Designability depends on the cluster’s geometry. At $N=8$, the cluster on the left has a minimal alphabet with only two species. The cluster on the right has only two alphabets, each of maximal size 8 (fourth row). At $N=9$ the minimal alphabet size is 3, and there are three different geometries that have such alphabets. (c) (Top panel) For each pairwise interaction between particle species (yellow and red), a valence can be set (see matrix). The valence can be controlled in different ways (see main text), for example, using DNA strands that are grafted on mobile linkers (top row). The valence limits the possible bonds in structures and allows control over reactions in bulk (white box). (Bottom panel) The top row shows two different types of time-dependent pairwise interactions, with time scale $\tau$. Templating reactions require both interactions that strengthen with time (left) and weaken with time (right). The bottom graphic shows a catalytic cycle involving a $N=7$ chiral cluster with the numbered particle species having the specified interaction matrices and valences toward monomers (center). Complex catalytic behavior, including self-replication, results from particles with valence of two (particle species 7) and variability in the time scale $\tau$ of strengthening and weakening interactions.

Figure 2

The equilibrium probabilities of clusters at $N=6$, 7, and 8 vary with the structure, even though all structures at a given $N$ have the same total depletion potential ([46]; [54]). For $N=6$, the less symmetric cluster, the polytetrahedron, occurs about 24 times as frequently as the more symmetric octahedron, even though both have $3N-6$ contacts. Similar variations in the probabilities for each structure arise at $N=7$ (five structures, each of which has $3N-6=15$ contacts) and $N=8$ (13 structures, each of which has $3N-6=18$ contacts). At both $N=7$ and $N=8$ probabilities for certain structures are grouped together. Eight high-magnification optical micrographs of colloidal clusters with $N=6$, 7, and 8 are shown on the periphery. From [46].

Figure 3

Absolute yield of self-assembly as a function of temperature, measured from simulations of a cluster with $N=8$ particles. Each data point is an ensemble average over 1000 simulations with different initial conditions. From time-averaged simulations we deduce that equilibrium assembly occurs for temperatures above $T/\epsilon \sim 0.1$. There are six alphabets for which the desired cluster geometry is the energetic ground state, and each is simulated independently. Within an alphabet each particle color represents a type of DNA coating, while the corresponding matrix shows the interactions among them, attractive (gray) or repulsive (white). For a particular alphabet size, the presence of attraction between far-away particles (“cross talk,” red matrix entries) introduces additional local minima and reduces the yield. The seventh curve (bottom) is the yield of the cluster using identical particles, where the wire diagram shows the cluster geometry. The cluster has a chiral partner that we also identify as the ground state.

Figure 4

Energy landscapes for the $N=8$ cluster shown in Fig. 3, designed using three different alphabets. Within an alphabet [(a), (b) or (c)], each particle color represents a type of DNA coating, while the corresponding matrix shows the interactions among them, attractive (gray) or repulsive (white). Only the lowest-energy local minima are shown, each missing one bond compared to the ground state (GS). #BB* is the minimal number of bonds that need to be broken for a transition to be possible. #PW is the number of distinct pathways by which the transition with fixed #BB* can be achieved. Although the size of the alphabet (that is, the specificity of the interactions) decreases from (a) to (c), (b) has the largest number of low-energy local minima, and the lowest yield. Note that #PW is quoted per GS, as the ground-state cluster and the local minima have chiral partners (not shown). A small fraction of the quoted number of pathways actually connect the local minimum to the chiral partner of the ground state.

Figure 5

Absolute yield as a function of temperature for the self-assembly of four complex structures: two bipyramids made of $N=44$ and $N=19$ different particle species, a linear chainlike structure made of $N=19$ different particle species and a Big Ben structure with $N=69$ different particles species. Each data point is an ensemble average over 100 simulations with different initial conditions. The significantly smaller yield of the chainlike structure compared to the bulky bipyramid is caused by the different number and energy of low-energy local minima that are created by local rearrangements of particles in either structure. Adapted from [80].

Figure 6

(a) Self-sustained reaction scheme for self-replication of a $N=7$ chiral cluster with a dimer catalyst. Each particle in these two parent clusters (stages I and V) can attach only one appropriately colored monomer, with complementary DNA coating (stages II and VI). The attached monomers can interact and form bonds (stages III and VII) until a certain number is reached, after which the attached monomers detach from the parent clusters (stages IV and VIII). The detached 7-monomer network folds into a chiral cluster replica and together with the detached dimer represents the new (complementary) parents (stages V and I). Because of the complementarity, the initial cluster can reproduce through a hypercycle (stages I–VIII). The complementary copy of the cluster is also its mirror image (chiral partner), which is expected owing to the geometry of templating from the surface; if the monomer network has too few bonds ($<11$) it can fold to have the chirality of the original cluster. (b) Interaction matrix between different particle species present in simulations. (c) The total number of chiral clusters as a function of time, as measured from DPD simulations. (d) Simulation snapshot showing eight replicas. One of the replicas is a local minimum. The attachment and detachment process for monomers requires time-dependent interactions and limited particle valence.

Figure 7

Emergence of catalytic cycles. (a) Snapshot of a simulation started with a single $N=7$ chiral cluster, where all generated clusters were allowed to template other clusters. The interaction matrix and valence rules used in this simulation are shown in Fig. 1. Adapted from [79]. (b) Family tree of all clusters in (a), where the valence-two particle is colored gray. Maximal size of a cluster templated here is $N=9$. (c) Example of a possible catalytic cycle among rigid clusters of sizes $N=7$ to 9. Arrows connect a cluster to structures it can template.