The self-assembly of complex structures is common in the natural world; for example, given the appropriate cellular environment, viral shells are spontaneously made out of proteins. We consider `self-assembly' to be the mechanism by which individual components and environmental conditions alone are sufficient to cause the spontaneous assembly of a given structure. Human assembly on the other hand is a step-by-step process, requiring the mechanical control of each element of fabrication. While human fabrication techniques are quite advanced, they start to become limited at the small, micron and sub-micron scale, where machine precision can become a limiting factor. For example, it is virtually impossible to avoid nanoscale defects in materials -- but the nanoscale properties of a material effect its function at the macroscale. Nature on the other hand, has elegantly solved this problem through the hierarchical fabrication of complex structures out of nanoscale building blocks. As one example, the sea sponge Euplectella sp. forms a macroscopic cylindrical square-lattice cagelike structure out of nano-scale silica spheres, which shows outstanding mechanical rigidity and stability -- overcoming the fragile nature of its constituent material, glass. The overarching question of self-assembly is, can we harness the natural assembly process so that `human-made objects' are made spontaneously? -- thereby expanding the realm of what we can make, and the ease with which we can make it.
Studies of self-assembly can be broken into 2 steps: (i) determining the set of all self-assemblable structures, and (ii) devising a method of getting any one structure out of the set to be the only one that forms. As a model system for self-assembly, Michael Brenner, Vinny Manoharan, and I have studied spherical colloidal particles. In this case, the set of all self-assemblable structures is the set of all possible sphere packings. Spherical colloidal particles are readily available, and conditions that cause them to bind to one another, and thus to self-assemble into packings, are well known. However, the set of all possible sphere packings was unkown. Thus, we developed an algorithm that uses graph theory and geometry to analytically derive a provably complete set of all sphere packings. Thus far, we have enumerated all packings of $n \leq 10$ particles, and are in the process of going to higher $n$. To enable the robust assembly of any one given packing, the spheres must be endowed with specific adhesion, restricting the allowed contact set. Such adhesion is experimentally accessible by, for example coating colloids with DNA or proteins. We have proven that there always exists an adhesive labeling that will cause any packing of $\geq 3n-6$ contacts to become a unique packing of $n$ particles -- thus leading to the robust self-assembly of such a structure.
The mathematical problems suggested by assembling finite sphere packings are modern variations on an old and elegant field. The perspective of sphere packings from the vantage point of self-assembly is a valuable one, and allows us to attack old problems in different ways. I discuss only a few such problems below:
The Erdos repeated distance problem (a.k.a unit distance problem) asks what is the maximum number of repeated (or unit) distances that can connect $n$ points in $d$ dimensions? This is a major unsolved problem in geometry - to this day only upper and lower limits are known, even in 2 and 3 dimensions. The solution to this problem in 3 dimensions, where the unit distance is also the minimum distance, corresponds to the maximum number of contacts possible in a packing of $n$ spheres. Can we solve this modified Erdos problem in 3 dimensions? From the set of sphere packings of $n \leq 10$, we know that for $n \leq 9$ spheres, the maximum number of repeated distances $= 3n - 6$. At $n = 10$, 2 packings arise that have $25 = 3n - 5$ contacts. Thus, the problem is solved for $n \leq 10$ spheres – can we extend this to derive a solution for general $n$?
Rigidity in 3 Dimensions: The onset of non-rigid packings satisfying minimal rigidity constraints ($\geq 3$ contacts per particle, $\geq 3n-6$ total contacts) occurs at $n = 9$. Why is this? How do such non-rigid packings grow with $n$? Is there a simple way to read off rigidity in 3 dimensions from an adjacency matrix? Do the automorphism groups or eigenvalues of such matrices have special properties that could identify them as rigid?
Non-identical Particles: Thus far, we have considered only systems of identical spherical particles. The dynamics change if particles have different shapes or sizes. For example, m & m's (i.e. ellipsoids) can form denser packings than spheres. How does the nature of packings change for ellipsoids, or for spheres with non-identical radii?
Perfect Crystals: Can we direct the self-assembly of a perfect crystal? Does an adhesive labeling exist that can take particles into the crystal phase without forming any defects in the crystal? Because we have shown that any packing of $\geq 3n-6$ contacts can be stabilized, this should in theory be possible if the colloids are initially disperse, as in a gas. However, beginning in the liquid phase, it is possible that equilibrating to a perfect crystal would take infinitely long, and thus be effectively inaccessible. In addition, boundary effects could become important. Furthermore, even if such a transition were possible for identical spheres -- if the colloids were not identical, would a perfect crystal be possible? How would this depend upon the distribution of radii?
Nanoscale Properties: The sub-micron characteristics of a material dictate its macroscale properties. Thus, by directing self-assembly at the micron and sub-micron scale, we have the potential to direct the self-assembly of objects and materials endowed with certain properties, such as extreme strength or flexibility. Given a particular attribute of interest - what should the nanoscale structure be? What is the best way to direct the self-assembly of this nanostructure and the ensuing macrostructure?
Mathematical Methods in Biology Show
Because biological systems have many components, it has become common to model them using large systems of coupled ordinary differential equations. However, there is as yet no simple way of determining how solutions to large systems of equations depend on their parameters. With Michael Brenner, we have shown that large biological signaling networks can be reduced to systems involving a few equations and effective parameters. The effective parameters lump the system's many components together, yielding a simplified system that contains within it information on all of the many components and that demonstrates which features of the signaling network are essential.
We are currently developing an algorithm for the reduction of such systems, so that this method can be easily implemented. I am also interested in determining whether it is possible to derive a finite set of reduced systems as a function of dimensionless parameters (ala low and high Reynolds number in fluid dynamics, albeit more complicated). Additionally, deriving when a reduced model can not be constructed as a function of dimensionless parameters and system type will be of interest.
I am also interested in combining this method with experimentation to investigate the following biological systems:
Cell Fates: How does a cell decide to differentiate, or in what to differentiate? How does it decide whether to divide, apoptose, or senesce? As just one example, the protein p53 is involved in apoptosis, cell division, and in cellular senescence. Depending on the conditions, this one protein is involved in leading a cell down any one of these fates. However, it is not understood exactly how this is controlled. Part of the difficulty in understanding this process lies in the fact that so many components are involved in these processes. It thus seems plausible that this model reduction method might allow us to parse through all the components, and to determine which ones are responsible for dictating a particular cell fate.
Organ Formation: Can a functional organ be made from its constitutive parts (e.g. cells, certain nutrients, etc.) in vitro? The growth of cartilage in the shape of a human ear on a mouse's back is a famous example of potentially growing an organ away from its normal environment. Scientists have begun to be able to grow blocks of a given cell type, such as kidney cells -- however, these blocks do not form a functional organ. What are the underlying processes that are missing in the lab setting, but exist within the organism that allow for organ formation?
Studies of aging have correlated several factors, primarily related to mild stress (such as caloric restriction), with lifespan extension. Proteins that mediate lifespan extension via these mechanisms, such as sirtuins, have also been identified. Additionally, proteins and enzymes, such as p53 and telomerase, involved in the mortality of cells have been identified, and correlations of the levels of these proteins with the age of mammals, including humans, have been documented. However, despite the wealth of correlations and manipulators of aging that have been determined, what aging is is still not understood.
My research on aging has been divided into two parts (i) establishing correlations between various proteins and aging, and (ii) defining what aging is.
Correlating Protein Expression with Aging: I developed a model of cell viability based on the ability of TRF2, a telomere related protein, to bind telomeres. Telomere length decreases with the replicative age of a cell, and the model's hypothesis (which is supported by preliminary experimental results) is that the binding of TRF2 to telomeres is length-dependent. Therefore, cells with shorter telomeres are less able to bind TRF2. If TRF2 is not bound to telomeres, the cell incorrectly recognizes these chromosome ends as double stranded breaks and attempts to rejoin them, leading to multicentromeric choromosomes. This in turn usually results in either cellular apoptosis or senescence (which occurs is dependent on whether the cell contains functional p53). The model is able to explain current experimental data on cell viability, and is thus one plausible explanation of cellular viability. However, the model contains 2 free parameters, and has yet to be experimentally verified. Additionally, even if correct, there are undoubtedly other independent determinants of cell viability. While cell viability is related to aging, exactly how it is related is unknown – efforts in constructing a definition of aging explore this relation. With David Sinclair and Fred Ausubel, I experimentally tested the theory of xenohormesis. Xenohormesis ('xeno' meaning cross and 'hormesis' meaning a favorable biological response to a low exposure of toxins or other stressors) is the theory that organisms feeding on a stressed plant will respond as if they were stressed themselves. Studies have shown that organisms subjected to mild stress, such as caloric restriction or heat shock, live longer. It has also been shown that sirtuins, which are NAD-dependent histone deacetylases, are involved in these pathways and responsible for the observed stress-dependent lifespan extension in lower organisms, such as yeast and c. Elegans. Sirtuin activating compounds (STACs), such as resveratrol, activate sirtuins, allowing the enzymes to operate at a higher rate. Thus, giving an organism STACs also results in its lifespan extension. Plants are the only organism that are capable of producing STACs, even though virtually every organism explored thus far contains sirtuins. Thus, xenohormesis predicts that animals feeding on stressed plants will live longer than those feeding on non-stressed plants because: plants will produce higher levels of STACs in response to stress, the organism will thus ingest more STACS, thereby activating its own sirtuins at a higher level, resulting in its increased lifespan. We have tested this theory using aphids as the model organism and Arabidopsis thaliana as the plant. The stress condition was high light.
Defining Aging: I have constructed a simple theory of organ and organismal aging based on the fact that a cell can do 1 of 4 things: (i) replicate, (ii) differentiate, (iii) die (either via apoptosis or necrosis), or (iv) senesce. The aging of organisms under different experimental conditions that alter its lifespan can be explained by this theory. More importantly, a simple definition of organismal aging arises: the aged state of an organ (and organism) is defined by i) the number of cells it contains, and ii) the distribution of states in those cells (whether the cell is a stem cell, a differentiated cell, or a senesced cell, and what generation it is in). Also, the theory yields predictions on how to manipulate the aged state of an organ and on how to prevent/eradicate cancer (modeled here simply as uncontrolled cellular growth).
- The Structure of Cadherins
- Spine Morphology in Neurons
- Multidimensional Secession