Autocatalytic Biochemical Networks and Their Fundamental Limits

In the present work, we study autocatalytic pathways which contain reactions that need the use of one of their own productions. These pathways are common in biology; one of the simplest and widely studied autocatalytic pathways is Glycolysis. This pathway produces energy by breaking down Glucose. It is shown that this pathway can be simplified as a network of three biochemical reactions. We first revisit some conditions on the underlying structure of the autocatalytic network, which guarantee the existence of fundamental limits on the output energy of such networks. Then we focus on autocatalytic pathways with several biochemical reactions. Our aim is to characterize the zero-dynamics for a class of autocatalytic networks and then study the fundamental limitations of feedback control laws, using their associated zerodynamics. For this aim, it is shown that the zero-dynamics of autocatalytic networks play an important role in studying the fundamental limits on performance. Zero-dynamics is defined as the dynamics of a system restricted to the control input and initial conditions such that the output of the system remains zero for all future time instances. We characterize the zero-dynamics for a class of unperturbed autocatalytic networks based on the structure of the original network. It is well-known that by knowing the zero-dynamics of a specific class of systems, one can obtain lower bounds on the best achievable performance (L2-norm of the output) for the system. For a specific class of autocatalytic networks, we characterize their zero-dynamics in terms of the state-space matrices of the underlying network. This can be utilized to quantify inherent fundamental limits on performance (the level of disturbance attenuation) for this class of network. In general, one should apply numerical algorithms to obtain such fundamental limits. We explain our method using a simple but illustrative example.