Abstract. Neural codes allow the brain to represent, process, and store information about the world.

Steady-state analysis of dynamical systems for biological networks gives rise to algebraic varieties in high-dimensional spaces whose study is of interest in their own right. We demonstrate this for the shuttle model of the Wnt signaling pathway. Here, the variety is described by a polynomial system in 19 unknowns and 36 parameters. It has degree 9 over the parameter space. This case study explores multistationarity, model comparison, dynamics within regions of the state space, identifiability, and parameter estimation, from a geometric point of view. We employ current methods from computational algebraic geometry, polyhedral geometry, and combinatorics.

Phylogenetic networks are becoming increasingly popular in phylogenetics since they have the ability to describe a wider range of evolutionary events than their tree counterparts. In this paper, we study Markov models on phylogenetic networks and their associated geometry. We restrict our attention to large-cycle networks, networks with a single undirected cycle of length at least four. Using tools from computational algebraic geometry, we show that the semi-directed network topology is generically identifiable for Jukes-Cantor large-cycle network models. arXiv:1706.03060v1 [q-bio.PE] 9 Jun 2017 do not assume any knowledge about which sites were produced by the same subtree of the network. The two-state Cavender-Farris-Neyman model may seem the more natural starting point for our exploration of network identifiabilty. However, as is evident from our computations in Proposition 4.7, the restricted coordinate space for this model makes it impossible to identify small networks from one another, our main strategy for eventually proving identifiability in the Jukes-Cantor case.Since the Jukes-Cantor model is time-reversible, the precise location of the root within the network will be unidentifiable from the distribution. However, we cannot simply study the unrooted topology of networks without orientation, since reticulation edges, edges directed into vertices of indegree two, play a special role defining the distribution. Thus, our results concern the identifiability of the semi-directed network topology, the unrooted, undirected network with distinguished reticulation edges. We will also restrict our attention to networks with only a single reticulation vertex which we call cycle-networks. We will refer to the set of all cycle-networks with cycle length greater than 4 as the class of large-cycle networks. The main result of this paper is the following theorem.Theorem 1.1. The semi-directed network topology parameter of large-cycle Jukes-Cantor network models is generically identifiable.Markov models on networks with a single reticulation vertex are very closely related to 2-tree mixture models but with some subtle differences that we discuss in Section 2.1. Using techniques from algebraic statistics, it is shown in [2] that the tree parameters of a 2tree Jukes-Cantor mixture are generically identifiable. Here we adopt a similar approach. We associate to each network N an algebraic variety V N that is the Zariski closure of the set of probability distributions attained by varying the numerical parameters in the model on N . We then study the associated ideals of the networks to find algebraic invariants that distinguish networks from one another. The two networks in Figure 1 demonstrate that the generic identifiability results for 2-tree mixtures do not apply for phylogenetic networks. These networks have different semi-directed network topologies and induce different multisets of embedded trees. Suprisingly, however, the algebraic variety for the network on the left is properly contained in that of the network on th...

This work focuses on the question of how identifiability of a mathematical model, that is, whether parameters can be recovered from data, is related to identifiability of its submodels. We look specifically at linear compartmental models and investigate when identifiability is preserved after adding or removing model components. In particular, we examine whether identifiability is preserved when an input, output, edge, or leak is added or deleted. Our approach, via differential algebra, is to analyze specific input-output equations of a model and the Jacobian of the associated coefficient map. We clarify a prior determinantal formula for these equations, and then use it to prove that, under some hypotheses, a model's input-output equations can be understood in terms of certain submodels we call "output-reachable". Our proofs use algebraic and combinatorial techniques.

While a diverse set of design strategies have produced various chemical tools for biomolecule labeling in aqueous media, the development of nonaqueous, biomolecule-compatible media for bioconjugation has significantly lagged behind. In this report, we demonstrate that an aprotic ionic liquid serves as a novel reaction solvent for protein bioconjugation without noticeable loss of the biomolecule functions. The ionic liquid bioconjugation approach led to discovery of a novel triphenylphosphine-mediated amine–azide coupling reaction that forges a stable tetrazene linkage on unprotected peptides and proteins. This strategy of using untraditional media would provide untapped opportunities for expanding the scope of chemical approaches for bioconjugation.