Switching and bistability has been recognized as an important feature of dynamical systems originating in Systems Biology. In the majority of cases bistability of the dynamical system is established numerically. However, parameter uncertainty is a predominant issue in Systems Biology : the dynamical systems consist of a large number of states and parameters, while measurement data are often very noisy and data points and repetitions are usually few. Hence techniques allowing the analytic computation of parameter vectors where a given system exhibits bistability are desirable. Here we present a new sufficient condition for a large class of mass action networks. This condition takes the form of linear inequality systems. And it is constructive in the sense that solutions to one of the inequality systems determine state and parameter vectors where the underlying mass action systems -- under explicitly stated genericity conditions -- undergo a saddle-node bifurcation  .
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) .
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Switching and bistability has been recognized as an important feature of dynamical systems originating in Systems Biology. In
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
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bistability of the dynamical system
, bistability
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However, parameter uncertainty is a predominant issue
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
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: the dynamical systems consist of a large number of states and parameters, while measurement data are often very noisy and data points and repetitions are usually few. Hence techniques allowing the analytic computation of parameter vectors where a given system exhibits bistability are desirable. Here we present a new sufficient condition for a large class of mass action networks. This condition takes the form of linear inequality systems. And it is constructive in the sense that solutions to one of the inequality systems determine state and parameter vectors where the underlying mass action systems
. 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
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explicitly stated
certain
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undergo
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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)