Switching in mass action networks based on linear inequalities

Many biochemical processes can successfully be described by dynamical systems allowing some form of switching when, depending on their initial conditions, solutions of the dynamical system end up in different regions of state space (associated with different biochemical functions). Switching is often realized by a bistable system (i.e. a dynamical system allowing two stable steady state solutions) and, in the majority of cases, bistability is established numerically. In our point of view this approach is too restrictive, as, one the one hand, due to predominant parameter uncertainty numerical methods are generally difficult to apply to realistic models originating in Systems Biology. And on the other hand switching already arises with the occurrence of a saddle type steady state (characterized by a Jacobian where exactly one Eigenvalue is positive and the remaining eigenvalues have negative real part). Consequently we derive conditions based on linear inequalities that allow the analytic computation of states and parameters where the Jacobian derived from a mass action network has a defective zero eigenvalue so that -- under certain genericity conditions -- a saddle-node bifurcation occurs. Our conditions are applicable to general mass action networks involving at least one conservation relation, however, they are only sufficient (as infeasibility of linear inequalities does not exclude defective zero eigenvalues).Comment: in revision SIAM Journal on Applied Dynamical System

N-site phosphorylation systems with 2N-1 steady states

Multisite protein phosphorylation plays a prominent role in intracellular processes like signal transduction, cell-cycle control and nuclear signal integration. Many proteins are phosphorylated in a sequential and distributive way at more than one phosphorylation site. Mathematical models of $n$-site sequential distributive phosphorylation are therefore studied frequently. In particular, in {\em Wang and Sontag, 2008,} it is shown that models of $n$-site sequential distributive phosphorylation admit at most $2n-1$ steady states. Wang and Sontag furthermore conjecture that for odd $n$, there are at most $n$ and that, for even $n$, there are at most $n+1$ steady states. This, however, is not true: building on earlier work in {\em Holstein et.al., 2013}, we present a scalar determining equation for multistationarity which will lead to parameter values where a $3$-site system has $5$ steady states and parameter values where a $4$-site system has $7$ steady states. Our results therefore are counterexamples to the conjecture of Wang and Sontag. We furthermore study the inherent geometric properties of multistationarity in $n$-site sequential distributive phosphorylation: the complete vector of steady state ratios is determined by the steady state ratios of free enzymes and unphosphorylated protein and there exists a linear relationship between steady state ratios of phosphorylated protein

Multistationarity in sequential distributed multisite phosphorylation networks

Multisite phosphorylation networks are encountered in many intracellular processes like signal transduction, cell-cycle control or nuclear signal integration. In this contribution networks describing the phosphorylation and dephosphorylation of a protein at $n$ sites in a sequential distributive mechanism are considered. Multistationarity (i.e.\ the existence of at least two positive steady state solutions of the associated polynomial dynamical system) has been analyzed and established in several contributions. It is, for example, known that there exist values for he rate constants where multistationarity occurs. However, nothing else is known about these rate constants. Here we present a sign condition that is necessary and sufficient for multistationarity in $n$-site sequential, distributive phosphorylation. We express this sign condition in terms of linear systems and show that solutions of these systems define rate constants where multistationarity is possible. We then present, for $n\geq 2$, a collection of {\em feasible} linear systems and hence give a new and independent proof that multistationarity is possible for $n\geq 2$. Moreover, our results allow to explicitly obtain values for the rate constants where multistationarity is possible. Hence we believe that, for the first time, a systematic exploration of the region in parameter space where multistationarity occurs has become possible.One consequence of our work is that, for any pair of steady states, the ratio of the steady state concentrations of kinase-substrate complexes equals that of phosphatase-substrate complexes

Ecological and evolutionary drivers of microbial community structure in termite guts

Presumably descending from subsocial cockroaches 150 million years ago, termites are an order of social insects that gained the ability to digest wood through the acquisition of cellulolytic flagellates. These eukaryotic protists fill up the bulk of the hindgut volume and are the major habitat of the prokaryotic community present in the digestive tract of lower termites. The complete loss of gut flagellates in the youngest termite family Termitidae, also called higher termites, led to an entirely prokaryotic gut microbiota as well as a substantial dietary diversification and enormous ecological success. While the subfamily Macrotermitinae established a symbiosis with wood-degrading fungi of the genus Termitomyces, other higher termites exploit diets with a higher degree of humification. Previous studies on the gut communities of termites have observed that while the gut microbiota of closely related hosts is very similar, those of more distantly related hosts are characterized by considerable differences in gut communities. Since these observations are based on highly limited samplings of hosts, it is uncertain if these differences reflect important evolutionary patterns. This dissertation includes studies examining the archaeal and bacterial diversity of the gut microbiota over a wide range of termites using high-throughput sequencing of the 16S rRNA genes. In comparison to the rather simple archaeal communities, which were mainly composed of methanogens, the bacterial gut microbiota were characterized by considerably higher diversity. At the phylum-level, Bacteroidetes, Firmicutes, Proteobacteria and Spirochaetes were ubiquitously distributed among the termites, albeit with differences in relative abundance. Other phyla, however, such as Elusimicrobia, Fibrobacteres and the candidate division TG3, occured only in certain host groups of termites. The distribution pattern of archaeal and bacterial lineages reflects both host phylogeny and differences in the digestive strategy of the host. Although several genus-level bacterial lineages showed a certain degree of host-specificity, phylogenetic analyses of the amplified rRNA genes showed that these bacterial lineages do not appear to be cospeciating with their hosts. The findings of studies included in this dissertation and other published studies were evaluated to identify potential drivers of community structure and other shaping mechanisms. Thus, gut community structure in termites is primarily shaped by habitat and niche selection. The stochastic element of these mechanisms, however, is strongly attenuated by proctodeal trophallaxis, which facilitates coevolution and might ultimately lead to cospeciation. While coevolution is likely true for many lineages and documented by host-specific microbial lineages, there is only little evidence of cospeciation in the gut microbiota of termites. If present, it is restricted almost exclusively to flagellates and their symbionts in lower termites. The higher wood-feeding termites have long been associated with a marked abundance of the phyla Fibrobacteres and cand. div. TG3. Although these phyla have been shown to be members of a specific cellulolytic community associated with wood particles in the hindguts of higher termites, their full functional potential still remains unknown. In order to elucidate the role of these organisms, a study in this dissertation carries out metagenomic analyses of various higher termites. In wood-feeding representatives, Fibrobacteres and cand. div. TG3 were in fact highly abundant, but only a few or no genes could be assigned to both groups by the usual database-dependent classification programs due to the lack of suitable genomes in these databases. In response, a new study was conceived to compensate this discrepancy. By further development of a new reference-independent method, over 30 population genomes of Fibrobacteres and cand. div. TG3 could be reconstructed from the metagenomic data sets. Subsequent comparative analysis revealed that organisms of both groups differ in their potential of wood degradation, but likely complement each other. Further analyses indicate that representatives of both groups might be able to fix nitrogen and respire under hypoxic conditions — two favourable adaptations to the unique termite gut environment

The development of functional inputs to a neural circuit : synaptic strength before and after the activity-dependent maturation of the retinogeniculate system

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Brain and Cognitive Sciences, 1999.Includes bibliographical references (leaves 181-193).by Carsten Dietrich Hohnke.Ph.D