Mathematics's role in the grand challenge of deciphering the molecular basis of life.

Biochemistry, the field that studies biomolecules and their relation to life, is a primary beneficiary of the flood of data that is emerging from the new technologies that allow us to observe biological systems at spatial and temporal scales never achieved before. It remains extraordinarily difficult to grasp the implications and to develop experimentally testable predictions from such complex data. Most of the data are fragmented and only pertain to either a static vision of biology, or provide information on dynamics over a limited time scale. However, the dynamics of biomolecular systems occurs over a large range of time scales. Thermal fluctuations for example occur in the femtosecond (fs)-to-picosecond (ps) time scale, while tumbling occurs in the nano second (ns) time range. Global molecular rearrangements such as those observed in allosteric regulation take place in the microsecond (μs) to millisecond (ms) time range while folding, binding, diffusion, motility and translocation befall in the ms-s-min time range. Macromolecular synthesis and supramolecular processes are even slower processes occurring in the seconds-to-hours time frame. Furthermore, even simple observations of biomolecular systems can require quantitative analyses that are beyond the repertoire of traditional methods available to experimentalists. The scarcity of comprehensive methods is even more acute with regard to answering fundamental questions of the chemistry and physics underlying the multitude of phenomena that define biomolecular systems and their interactions. The current views of the relationship between biomolecular structure and function for example remain fragmented. We know of their sequences, more and more about their structures, and we have information on their biological activities, but we have difficulties connecting these dots into a knowledgeable whole. Thus, biochemistry requires new theoretical and computational approaches toward organizing data into quantitative models. Mathematics (broadly defined) is well positioned to play a major role in these efforts, by working collaboratively with bench biologists, chemists and physicists. Mathematics has always played an important role in biology, including biochemistry. There is indeed a field of research named “mathematical biology” whose aims are to provide quantitative representations, treatments and modeling of biological processes, using a variety of applied mathematical techniques and tools. By describing biomolecular systems in a quantitative manner, their behavior can be better simulated, and hence properties can be predicted that might not be accessible to the experimenter. Mathematical disciplines such as probability theory, statistics, combinatorics, dynamical systems, and the study of differential equation have traditionally provided the tools for such quantitative analyses. The ongoing transformation of biology into a quantitative science is increasing the need, however, for new tools and new theories. As a consequence, I expect that mathematical areas such as calculus, linear algebra, abstract algebra, graph theory, algebraic geometry, topology, and coding theory will play an increasing role in the new era for biology. To get a better understanding of the challenges that lie ahead of us, it is worth considering a simple example with significant consequences for human health. Dengue virus is a positive sense RNA virus responsible for dengue fever, a tropical infectious disease whose incidence has increased drastically over the last decades. There are currently no prophylactic treatments against dengue fever, with the exception of eliminating the vector mosquitoes. The genome of dengue virus encodes for 10 different proteins. A perhaps surprising idea that has crystallized from years of studies of dengue virus is that its biology is deeply encoded in the dynamics of these proteins. For example, the envelope protein E on the surface of dengue viruses undergoes a change in conformation upon entering the endosome following the infection of a cell. This structural change is ultimately responsible for the release of the virus genome in the cell (Bressanelli et al., 2004; Modis et al., 2004). The surface of dengue virus is also known to be dynamic (Lok et al., 2008). An antibody that locks the virus surface in a static state blocks infection by preventing attachment of dengue virus to cells (Teoh et al., 2012). These are all key indicators that the dynamic movement of dengue surface proteins is important for cell attachment and ultimately its virulence. Our experimental knowledge of this dynamics that occurs at different time scales, however, is limited. How are structure and dynamics connected for the dengue virus? How do we connect dynamics to the biology of viral entry into cells? What is the connection between dynamics and to the inhibition of dengue virus with antibodies? These are key questions that are vital to the design of vaccines as well as to the identification of prophylactic antibodies. Deciphering the relationships between the structure, function, and dynamics of biomolecules is the grand challenge faced by biochemists in their attempt to unravel the mysteries of life. There are many facets of this challenge that remain unsolved. Of particular interest, the subjects of biomolecular structure prediction and folding remain key challenges for the community of computational and theoretical structural biologists. Interestingly, these areas define a treasure trove of problems where mathematics can prove useful. In the following two sections, I outline some of the mathematical challenges featured in these research areas. Hopefully some of these challenges will be met and discussed in the new Frontiers section, “Mathematics and Biomolecules.”