ABSTRACT
Nonlinearity appears to be a fundamental property of biological systems. One reason for this may be the inherent complexity of biology — in the physical world, linear equations such as Newton’s, Maxwell’s and Schroedinger’s are immensely successful descriptions of reality, but they are essentially equations of forces in a vacuum. Nonlinearity is fundamental in generating qualitative structural changes in complex phenomenon such as the transition from laminar to turbulent flow, or in phase changes from gas to liquid to solid. Whenever there are phase changes, whenever structure arises, nonlinear dynamics are often responsible, and the very fact that biological phenomena have for many years been successfully described in qualitative terms indicates the importance of nonlinearity in biological systems. As argued in [1], if it were not for nonlinearity, we would all be quivering jellies! More concretely, even the briefest consideration of the dynamics which arise
from the biochemical reaction kinetics underpinning almost all cellular processes, [2], reveals the ubiquity of nonlinear phenomena. The fundamental law of mass action states that when two molecules A and B react upon collision with each other to form a product C
A + B k→ C (3.1)
the rate of the reaction is proportional to the number of collisions per unit time between the two reactants and the probability that the collision occurs with sufficient energy to overcome the free energy of activation of the reaction. Clearly, the corresponding differential equation
dC
dt = kAB (3.2)
where k is the temperature dependent reaction rate constant, is nonlinear. In enzymatic reactions, proteins called enzymes catalyse (i.e. increase the rate of) the reaction by lowering the free energy of activation. This situation may be represented by the Michaelis-Menten model, which describes a two-step process whereby an enzyme E first combines with a substrate S to form a
in
form the product P
S + E k1−⇀↽− k2
C k3→ P + E (3.3)
The corresponding differential equation relating the rate of formation of the product to the concentrations of available substrate and enzyme is again nonlinear
dP
dt =
VmaxS
Km + S (3.4)
where the equilibrium constantKm = (k2+k3)/k1 and the maximum reaction velocity Vmax = k3E. Cooperativity effects, where the binding of one substrate molecule to the enzyme affects the binding of subsequent ones, serve to further increase the nonlinearity of the underlying kinetics. In general, for n substrate molecules with n equilibrium constants Km1 through Kmn, the rate of reaction is given by the Hill equation
dP
dt =
VmaxS n
Knh + S n
(3.5)
where Knh = ∏n
i=1Kmn. Nonlinear Michaelis-Menten and Hill-type functions are also ubiquitous in
higher-level models of cellular signal transduction pathways and transcriptional regulation networks. In transcriptional regulatory networks, for example, transcription and translation may be considered as dynamical processes, in which the production of mRNAs depends on the concentrations of protein transcription factors (TFs) and the production of proteins depends on the concentrations of mRNAs. Equations describing the dynamics of transcription and translation, [3], can then be written as
dmi dt
= gi(p)− kgimi dpi dt
= kimi − kpi pi respectively, where mi and pi denote mRNA and protein concentrations and kgi and k
p i are the degradation rates of mRNA i and protein i. The function
gi describes how TFs regulate the transcription of gene i, and experimental evidence suggests that the response of mRNA to TF concentrations has a nonlinear Hill curve form [4]. Thus, the regulation function of transcription factor pj on its target gene i can be described by
k hij ij + p
(3.6)
for the activation case and
k hij ij + p
(3.7)
i maximum rate of transcription of i, kij is the concentration of protein pj at which gene i reaches half of its maximum transcription rate and hij is a steepness parameter describing the shape of the nonlinear sigmoid responses. Finally, nonlinear dynamics are also ubiquitous at the inter-cellular level,
where linear models obviously cannot capture the saturation effects arising from limitations on the number of cells which can exist in a medium or organism. In fact, interactions between different types of cells often display highly nonlinear dynamics — consider, for example, a recently proposed (and validated) model of tumour-immune cell interactions, [5], which gives the relationships between tumour cells T , Natural Killer (NK) cells N and tumourspecific CD8+ T-cells L as,
dT
dt = aT (1− bT )− cNT −D
dN
dt = σ − fN + gT
h+ T 2 N − pNT
dL
dt = −mL+ jD
k +D2 L− qLT + rNT
D = d (L/T )γ
s+ (L/T )γ T
Again, the highly nonlinear nature of the dynamics governing the interactions between the different cell types is strikingly apparent in the above equations. For the reasons discussed above, the mathematical models which have been
developed to describe the dynamics of intra-and inter-cellular networks are typically composed of sets of nonlinear differential equations, i.e. nonlinear dynamical systems. The study of such systems is by now a mature and extensive field of research in its own right (see, for example, [6]) and so we will not attempt to provide a complete treatment here. Instead, and in keeping with the aims of this book, we will focus on certain aspects of nonlinear systems and control theory which have particular applicability to the study of biological systems.
