## ABSTRACT

Profound study of nature is the most fertile source of mathematical discoveries.

Fourier, Analytical Theory of Heat

One does not have to understand the science of optics to appreciate the beauty of a rainbow; but it helps. In the same way, one need not master the mathematics of self-assembly in order to appreciate the power of the concept; but here too, it helps. In the ﬁrst two parts of this book we focused on descriptions of self-assembling systems. At times, we made use of mathematics, but ultimately our focus was on experiment rather than theory. In this chapter, we shift our focus and examine the various theoretical approaches to understanding the phenomenon of self-assembly. There are as many diﬀerent approaches to mathematically modelling self-

assembly as there are examples of physical self-assembling systems. In the end, the type of model one constructs depends upon the type of question one wishes to answer. These questions can vary wildly. At one end of the spectrum, we have models built to illuminate the behavior of one speciﬁc selfassembling system. Such a model can have great utility. If accurate, it can help reduce the number of costly or time consuming experiments one needs to conduct. It can clarify the role of various parameters in the system and give a picture of parameter space that might otherwise be inaccessible. At its best, it can clarify a complex situation, help guide experiment, and identify new experimental regimes to be explored. At the other end of the spectrum we have abstract models of the phenomenon of self-assembly. These models are usually divorced from any particular experimental system; rather they seek to capture the behavior of some large class of self-assembling systems. These models too, can have great utility. At their best, they can help us answer “What is possible?” types of questions. Is it possible to self-assemble a Sierpinski Gasket in a system containing only two tile types? Is it possible

to self-assemble a cell given inﬁnitely many tile types? These are the types of questions that abstract models are best at answering. However, there is no hard and fast boundary between these types of models.

Models of a particular physical system are often found to apply to other systems, systems that at ﬁrst might seem unrelated. These models are perhaps more abstract than we initially thought. Abstract models take their inspiration from physical systems and in seeking to capture general principles, often end up capturing real behavior remarkably well. At times, a model that initially seemed abstract may end up being physically realizable, and end up showing us a new route to self-assembly. Nor is there any mathematical distinction between these types of models.

The equations of continuum mechanics can help us develop a detailed description of the shape of a meniscus, but they can also be implemented in a computer, governing the behavior of ﬁctitious particles that have no counterpart in the real world. Seemingly pure branches of mathematics, such as graph theory, which lends itself nicely to several abstract approaches, also lends itself nicely to robotic control schemes for real world engineered particles. Similarly, computer simulation plays an important role in the analysis of every kind of model. Both physically driven models and abstract models have a tendency to become analytically intractable. In both cases, numerical simulation becomes a necessity. Nonetheless, for clarity in the discussion, we will make a distinction be-

tween these two types of models. We’ll divide this chapter into two main sections. In the ﬁrst, Physical Models, we’ll describe approaches that stay close to one physical system or some small subclass of physical systems. In the second, Abstract Models, we’ll examine approaches to “What is possible?” type questions. In Section 9.2, Physical Models, we begin with a mathematical model of

the structured surfaces discussed in Chapter 6. This model asks the question: What can be accomplished if an electric ﬁeld is used to manipulate the minimal energy surfaces of Chapter 6? This model is very much at the “single experimental system” end of the spectrum. Through this model we’ll see how key parameters in a problem may be identiﬁed and how a model can help us understand parameter space and suggest experimental directions. Next, we’ll examine a model that attempts to explain why the helix is such a familiar structural motif in nature. In contrast to the structured surface model, this model focuses on a class of self-assembling systems rather than on a single experimental setup. We’ll see how such a model can be useful, both to give insight into a broad problem, and to actually predict experimental results. For our third model, we’ll return to the ﬁrst system we discussed in Chapter 6: the self-assembling tile system of Hosokawa et al. The model we’ll discuss is drawn from their original paper [62] describing their experimental and theoretical results. We’ll see how a model inspired by chemical reaction kinetics can capture the behavior of a tile based self-assembling system. Finally, in this section, we’ll discuss the so-called waterbug model, due to Eric Klavins.

This ﬁnal model is again unattached to any particular physical system, but is inspired by a class of such systems. With this model, we’ll see how theory can aid in the design of physical systems. In Section 9.3, Abstract Models, we focus on three abstract approaches to

modelling self-assembly. The notion of a conformational switch is the focal point of the ﬁrst of these models. We encountered conformational switching in Chapter 3 when we discussed the tobacco mosaic virus. We also encountered this notion when we discussed proteins and again in part two of this book in the context of several diﬀerent engineered systems. The model of conformational switching presented in this section attempts to characterize the power of a conformational switch to encode for a given assembly sequence. The second model we consider is based on the notion of a graph grammar. This model generalizes the conformational switch model and within the context of the model is able to provide a constructive solution to the backward problem of self-assembly. The ﬁnal model we consider is the Tile Assembly Model. This important model provides the link connecting self-assembly and computation. We’ll see how this model has been used to explore the question of complexity of a self-assembling system and how this model provides a promising route to programmed self-assembly. One ﬁnal note before we begin – to understand the details of every model

discussed in this chapter requires a broad mathematical background. Here, we won’t focus on these details. Rather, we’ll attempt to provide a sense of the thinking behind the model, the questions it seeks to address, and the importance of the answers to those questions. Further, where it seems most appropriate, we’ll ﬁll in the mathematical background needed to understand the basics of the model. However, this may not always be enough. If you ﬁnd the details of a particular model in this chapter to be confusing or inaccessible, skip them. You should still be able to get a sense of the model. If you still ﬁnd a particular model to be heavy going, skip it entirely. The subsections in this chapter are mostly independent.1 I encourage you to ﬁnd a modelling approach and a set of questions that excites you, and to continue from there.