
MBW:Biological Clocks and SwitchesFrom MathBioArticle Review by William Dowling and Toni Klopfenstein
ContentsSummary
Historical ContextAlfred Lotka and Vito Volterra, a physical chemist and mathematician respectively, were some of the first to consider and study biological interactions as dynamical systems. In particular, they studied the pair of nonlinear ordinary differential equations of the form: dx/dt=f(x,y) , dy/dt=g(x,y) where the variables x and y are the timedependent state of a chemical/biological system. These equations were then used to model predatorprey interactions, of the form: dx/dt=axbxy , dy/dt=cxydy These equations have nested periodic solutions (oscillations or ‘clocks’), determined by the following conservation of energy condition: E(x,y)=a ln(y)by+d ln(x)cx=constant The predatorprey model is modified into the following form in order to simulate two competing species: dx/dt=ax(Kx)bxy , dy/dt=cy(Ly)dxy
This system gives two steadystate solutions (a switch) at: (x=K,y=0) and (x=0,y=L) and an unstable steadystate at: (x=c (bLaK)/(bdac),y=a (dKcL)/(bdac)) if the condition: c/d<K/L<b/a is satisfied. Though Lotka and Volterra only modeled population dynamics with these systems, they can easily be applied to reactions of biochemicals within cells.
Because of the lack of information about biochemistry available before the 1950’s, the possibility of dynamic substrate conversion reactions was given little thought. In the 1950’s and 60’s, a number of examples of biochemical switches and clocks were recognized in the field of molecular biology, including periodic enzyme synthesis, and oscillations in enzymecatalyzed reactions. One example of chemical reactions with the presence of these ‘switches’ and ‘clocks’ is the “Brusselator’ equation: dx/dt=abx+cx^2 ydx , dy/dt=bxcx^2 yey as described by Prigogine and Lefever in 1968. The Brusselator is a dynamical model of a chemical reaction network which features oscillations, bistability, pattern formation and travelling waves. It is based upon the following ternary autocatalytic reaction: Y + 2X 3X. Depending on the selection of e=0 or e≠0, the Brusselator results in either a unique steady state, shown in figure 1 or bistability, shown in figure 2^{[1]}.
(x=a/d,y=bd/ac) and is unstable for: d^3bd^2+ca^2<0
Genetic regulatory systems have positive and negative feedbacks and were among the dynamic biological systems to be modeled in this period (To see another example of modeled gene regulation, check out Modelling the Tryptophan Operon). In particular, Griffith (1968) provided mathematical model for a positive feedback system, a negative feedback system, and a negative feedback system with n components. In his equations, x is a measure of mRNA and y is a measure of protein. The positive feedback system is described by the following system: dx/dt=(a+y^2)/(K^2+y^2 )bx,dy/dt=cxdy This system features bistability of gene expression dependent on the parameter d, which represents demand for the protein. Specifically, the steady state of the system changes from low to high expression at a critical value of d. Griffith modified the positive feedback system equations to model a negative feedback system as follows: dx/dt=a/(K^p+y^p )bx,dy/dt=cxdy Interestingly, this system features neither periodic solutions nor bistability, but he proved that the extension of the above system for n different components can have limit cycle oscillations if certain conditions are met. The corresponding equations are as follows: (dx_1)/dt=a_0/(K^p+x_n^p )b_1 x_1 (dx_2)/dt=a_1 x_1b_2 x_2 (dx_n)/dt=a_(n1) x_(n1)b_n x_n Limit cycle oscillations arise when the exponent p of the nonlinear terms representing the feedback control satisfies the condition: p>〖〖(sec〗〖π/n〗)〗^n. Griffith’s work had a significant impact on the development of further models of genetic regulatory systems. Although the mathematical theory of switches and clocks was moving forward during the 1960’s and 1970’s, molecular biology’s understanding of the physical mechanisms of most reallife examples was lacking. With a few notable exceptions( glycolytic oscillations, cyclic AMP signaling, calcium oscillations and waves), the genes and proteins were essentially unknown and therefore not yet accessible to modeling. Because of the lack of information existing for many biological systems such as circadian rhythms, fruit fly development, coordination of cell division or chemotatic response of motile bacteria, the progress of modeling techniques stagnated in the 1980’s until breakthroughs in the field of molecular genetics at the end of the decade. In the 1990’s there were a number of successes in the field, and by 2000, molecular cell biology was experiencing a noticeable shift towards quantitative modeling, thanks to a large increase in support from several government agencies. Modern DevelopmentsSince 2000, many advances have been made in the field of molecular biological modeling and several institutes, centers and departments have been founded, as well as many new scientific journals. One major development was the design of artificial genetic networks by collaborators at Boston College and Princeton University which worked as toggle switches and oscillators. The computer models which were used to develop the circuits were similar to Griffith’s work. The models correctly predicted ranges for rate constants for synthesis and degradation of proteins. Thanks to work such as this, a new area of technology has developed known as synthetic biology. Advances have also been made in the field of network topology, including the work of Conrad, which used input pulse trains as resonators that were dependent on the frequency of pulses, and as integrators that were dependent on the number of pulses. This work led to the realization that topology is not as vital as the rate constant values are to functionality, and both characteristics must be considered when designing gene regulatory networks. Alongside the other expanses in molecular modeling, major improvements have led to a better comprehension of the molecular basis of circadian rhythms. Due to the diverse nature of circadian rhythms between species, the new models must be specified to each species, as well as allow for the major differences in the rhythms of cyanobacteria. Work has also been done on the molecular machinery controlling cell growth and division in mammalian and yeast cells, as well as αproteobacteria. Signaling networks, a major part of the workshop, have also led to new models and realizations, including the importance of feedback and feedforward signals to survival, behavior and development of cells. Discrete network and stochastic process models have also been used recently to examine cell division effects. ApplicationsOne specific application of modeling biological clocks and switches is the development of sleep regulation models, where two competing processes interact in an oscillatory manner, allowing the body to go through multiple cycles of different types of sleep. For more information on modelling sleep regulation, please see APPM4390:Modelling Sleep Regulation Current Models AvailableMany sophisticated tools are currently available to researchers, but the computational challenges that remain are great, and there is still much work needed. Below are some links to the tools accessible online.
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