Combinatorial Complexity
For more than a century, rate equations have been used to describe systems by posing questions in the following form: “How much of a molecule in some state X exists at time T?” This is the result of the population-based framework that is required when invoking the law of mass-action. Biochemists and biophysicists have sought to use these models to help understand a variety of processes, from gene transcription, to oxygen delivery to tissues, to cell signal transduction.
The molecules that are modeled in these processes exist in various states, and each state is typically represented by an individual data element. For example, a molecule bound to a substrate exists in a different state than a molecule of the same type in its unbound form or bound to a different substrate. Furthermore, macromolecules in which multiple subunits act as effectors of other subunits are often described as a combination of the individual states of the subunits, such that the number of states of the macromolecule is equal to:
(possible states ofsubunit1)×(possible states ofsubunit2). . .×(possible states ofsubunitn).
Thus, a homo-oligomer can to be described byabdata elements, whereaequals
the number of states in which a single subunit can exist, andbequals the number of subunits in a macromolecule. One example of this is cooperative binding, where inter-subunit interactions allow for increasing binding affinity upon subsequent binding events.
Modelers have grown accustomed to building models in which the number of data elements (i.e., the model’s data-space) equals the number of states of all molecules modeled in the system (i.e., the model’s state-space). Following this dogma, as the number of subunits in an oligomer increases, the number of data elements needed to describe its states increases exponentially. This is sometimes referred to as a combinatorial explosion (ab). In this work, we present a framework in which we are able to represent the full state-space of approximately ab states with a significantly smaller data-space, a×b. In the following chapters we will show that the combinatorial complexity is a function of the theory of mass action rather than a truth of real chemical reaction networks.
As an aside: It is more impressive to model chemical reaction networks with- out defining reactants, products, effectors, and reactions. By employing reactive molecular dynamics force-fields such as Reaxff (van Duin et al. [2001]), one can model complex biochemical systems without predefining many of the features re- quired in standard or modified models of chemical reaction networks. Code for optimization of Reaxff will be provided at the end of the second part of this thesis.
Chapter 3
Quantal Effectors on Lattices
In this chapter, we introduce the terminology used to describe molecular ele- ments that exhibit explicitly non-ideal behavior. We define these terms here so that we might describe the quantal effects that we see in future chapters. We will in- troduce quantal effectors that behave in an explicitly discrete manner. We will also discuss their early description as cooperative effectors and how they were treated in the mass-action framework. Next we will touch on statistical mechanics, specif- ically lattice theory as it has been applied to hemoglobin to model cooperative effects.
Following this discussion of discretization of hemoglobin subunits to points on a lattice, we will discuss how these ideas have been re-framed and extended by rule-based modelers. Finally, we will attempt to unify the insights made toward the understanding of hemoglobin with a general theme that we have revisited: that we do not have an appropriate language to describe molecular elements that affect reactions in a discrete manner, but are neither reactants nor products.
3.1
Introducing Quantal Effectors
Cofactors, promoters, and non-competitive inhibitors can and should be con- sidered “effectors” of reactions. We have excluded competitive inhibitors from this list because they directly interact with the binding site of at least one of the reacting molecules and, therefore, could play a very different role than effectors. A very important subclass of effectors are the subunits/domains of molecules and macromolecules that affect the reactivity of other subunits/domains. We will refer to these as “quantal effectors”.
Quantal effectors are molecular entities that are explicitly non-ideal. Naming these elements quantal effectors gives the user insight into the quantal behavior that may become apparent under certain conditions. An alternative name that is equally or more apt is non-ideal effector. Such a term is important because it is not domain-specific, and it provides insight into general truths. Domain-specific terms—such as lattices, agents, and rules—have specific meanings that are difficult to distinguish from the methods with which they are associated.
Since the early 20th century, we have been aware of complex reaction dynam- ics shown by molecular complexes. In 1925, Adair showed that hemoglobin has multiple binding sites for oxygen and that these sites potentiate one another when bound. While the reacting dynamics of hemoglobin can be coaxed into a mass- action type of model, it is non-ideal in that hemoglobin subunits autocatalyze one another in a non-ideal manner that is only partially governed by mass-action, as quantal effectors.