Mathematical Biology

Motivated by recent work with biologists, I will showcase some mathematical results on Turing instabilities in complex domains. This is scientifically related to understanding developmental tuning in a variety of settings such as mouse whiskers, human fingerprints, bat teeth, and more generally pattern formation on multiple scales and evolving domains. Some of these problems are natural extensions of classical reaction-diffusion models, amenable to standard linear stability analysis, whereas others require the development of new tools and approaches. These approaches also help close the vast gap between the simple theory of diffusion-driven pattern formation, and the messy reality of biological development, though there is still much work to be done in validating even complex theories against the rich pattern dynamics observed in nature. I will emphasize throughout the role that Turing's 1952 paper had in these developments, and how much of our modern progress (and difficulties) were predicted in this paper. I will close by discussing a range of open questions, many of which fall well beyond the extensions I will discuss, but at least some of which were known to Turing.

The plasma membrane contains a wide array of glycans and glycolipids, many of which are capped by sialic acids (also called neuraminic acid). As a result, sialic acids are front-line mediators of interactions between the extracellular surface and the external environment. Examples include host-pathogen interactions (e.g. influenza) and the recognition of host cells by leukocytes (white blood cells). Thus, the composition of sialosides in the membrane can influence receptor-receptor interactions critical to immunity and cellular function. Our group is investigating the influence of sialic acid on the function of adhesion and immune receptors through the development of tools that alter catabolism of membrane sialosides. The human neuraminidases (NEU) are a family of four isoenzymes (NEU1, NEU2, NEU3, and NEU4) which have a range of substrate preferences as well as cellular and tissue localization. Our group has developed a panel of selective inhibitors, many with nanomolar potency, are being used to investigate how degradation of sialosides influences the function of cellular receptors. We use fluorescence microscopy to measure the size of receptor clusters and lateral mobility of receptors. These biophysical methods provide critical insight into the influence of NEU activity on membrane receptor organization. We have examined the role of NEU enzymes on the function and organization of leukocyte adhesion receptors. We find that specific NEU enzymes can modulate integrin adhesion and affect leukocyte transmigration. In related work, we have examined the influence of synthetic glycoconjugates and inhibitors of NEU on the organization of the CD22 receptor of B cells. We propose that understanding the specific roles of NEU isoenzymes will identify new therapeutic strategies for autoimmunity, inflammation, and cancer.

Amyotrophic lateral sclerosis (ALS) is a fatal neurodegenerative disease primarily impacting motor neurons. Mutations in superoxide dismutase 1 (SOD1) are the second most common cause of familial ALS. Several of these mutations lead to misfolding or toxic gain of function in the SOD1 protein. Recently, we reported that misfolded SOD1 interacts with TNF receptor-associated factor 6 (TRAF6) in the SOD1-G93A rat model of ALS. Further, we showed in cultured cells that several mutant SOD1 proteins, but not wild type SOD1 protein, interact with TRAF6 via the MATH domain. Here, we sought to uncover the structural details of this interaction through molecular dynamics (MD) simulations of a dimeric model system, coarse grained using the AWSEM force field. We used direct MD simulations to identify buried residues, and predict binding poses by clustering frames from the trajectories. Metadynamics simulations were also used to deduce preferred binding regions on the protein surfaces from the potential of the mean force in orientation space. Well-folded SOD1 was found to bind TRAF6 via co-option of its native homodimer interface. However, if loops IV and VII of SOD1 were disordered, as typically occurs in the absence of stabilizing Zn2+ ion binding, these disordered loops now participated in novel interactions with TRAF6. On TRAF6, multiple interaction hot-spots were distributed around the equatorial region of the MATH domain beta barrel. Expression of TRAF6 variants with mutations in this region in cultured cells demonstrated that TRAF6 residue T475 facilitates interaction with different SOD1 mutants. These findings contribute to our understanding of the disease mechanism and uncover potential targets for the development of therapeutics.

Growing plant shoots exhibit circumnutations, namely, oscillations that draw three-dimensional trajectories, whose projections on the horizontal plane generate pendular, elliptical, or circular orbits. A large body of literature has followed the seminal work by Charles Darwin in 1880, but the nature of this phenomena is still uncertain and a long-lasting debate produced three main theories: the endogenous oscillator, the exogenous feedback oscillator, and the two-oscillator model. After briefly reviewing the three existing hypotheses, I will discuss a possible interpretation of these spontaneous oscillations as a Hopf-like bifurcation in a growing morphoelastic rod.

The principal function of the nucleus is to facilitate storage, retrieval, and maintenance of the genetic information encoded into DNA and RNA sequences. A unique feature of nucleoplasm—the fluid of the nucleus—is that it contains chromatin (DNA) and RNA.

In contrast to other important biological polymer hydrogels, such as mucus and extracellular matrix, the nucleic acid polymers have a sequence. Recent experiments have shown that during the growth phase of the cell cycle, chromatin condenses in a sequence specific manner into regions within the nucleoplasm, possibly so that functionally related genes are grouped together spatially even though they might be far apart in terms of sequence distance.

At the same time, we are becoming increasingly aware of the role of liquid-liquid phase separation (LLPS) in cellular processes in the nucleus and the cytoplasm. Complex molecular interactions over a wide range of timescales can cause large biopolymers (RNA, protein, etc) to phase separate from the surrounding nucleoplasm into distinct biocondensates (spherical droplets in the simplest cases).

I will discuss recent work modelling the role of nuclear biocondensates in neurodegenerative disease and several ongoing projects related to
modelling and microscopy image analysis.