John W. Cain

The whale shark genome reveals how genomic and physiological properties scale with body size

The endangered whale shark (Rhincodon typus) is the largest fish on Earth and is a long-lived member of the ancient Elasmobranchii clade. To characterize the relationship between genome features and biological traits, we sequenced and assembled the genome of the whale shark and compared its genomic and physiological features to those of 81 animals and yeast. We examined scaling relationships between body size, temperature, metabolic rates, and genomic features and found both general correlations across the animal kingdom and features specific to the whale shark genome. Among animals, increased lifespan is positively correlated to body size and metabolic rate. Several genomic features also significantly correlated with body size, including intron and gene length. Our large-scale comparative genomic analysis uncovered general features of metazoan genome architecture: GC content and codon adaptation index are negatively correlated, and neural connectivity genes are longer than average genes in most genomes. Focusing on the whale shark genome, we identified multiple features that significantly correlate with lifespan. Among these were very long gene length, due to large introns highly enriched in repetitive elements such as CR1-like LINEs, and considerably longer neural genes of several types, including connectivity, activity, and neurodegeneration genes. The whale shark’s genome had an expansion of gene families related to fatty acid metabolism and neurogenesis, with the slowest evolutionary rate observed in vertebrates to date. Our comparative genomics approach uncovered multiple genetic features associated with body size, metabolic rate, and lifespan, and showed that the whale shark is a promising model for studies of neural architecture and lifespan.

Nonlinear Systems: Global Theory

Theorem 3.2.1 guarantees that a solution to the initial value problem exists for what might be an extremely short time.

Nondimensionalization and Scaling

The preceding chapter had some pretty heavy analysis, and the next has even more. In what may be welcome relief, the present chapter pushes in an orthogonal direction: it focuses on nondimensionalization and scaling, which are techniques for simplifying ODEs that arise in applications.

Examples of Global Bifurcation

The main goal of this chapter is to give examples of five different types of global bifurcation. While the local bifurcations of the previous chapter were associated with stability changes in an equilibrium, the bifurcations in this chapter are associated with stability changes in a periodic solution. We make no pretense of completeness. It is not remotely possible to classify all possible global bifurcations. We introduce each type of bifurcation with an academic example that may be handled analytically, but for the more interesting examples that follow, we will rely heavily on computations (which we invite you to check) and on intuitive arguments. Even in cases for which proofs are available, we may omit them because we find them unrewarding. Consequently, this chapter is much less dense than the preceding one.

Linear Systems with Constant Coefficients

The bulk of this chapter is devoted to homogeneous linear systems of ODEs with real constant coefficients. This means systems of the form n n

Trajectories Near Equilibria

displaystyle{ begin{array}{ccc} x_{1}^{{prime}}& =& a_{11}x_{1} + a_{12}x_{2} +ldots +a_{1d}x_{d}, x_{2}^{{prime}}& =& a_{21}x_{1} + a_{22}x_{2} +ldots +a_{2d}x_{d}, vdots&vdots&vdots x_{d}^{{prime}}& =&a_{d1}x_{1} + a_{d2}x_{2} +ldots +a_{dd}x_{d}.end{array} }

Oscillations in ODEs

n n(2.1) n n(From now on, we shall let d be the dimension of our systems, so that the index n is available for other uses.) The written-out system (2.1) is awkward to read or write, and we shall normally use the vastly more compact linear-algebra notation n n