Homo-Oligomers in Signal Transduction: Dynamics, Homeostasis, Ultrasensitivity, Bistability

Homo-oligomerisation of proteins is a ubiquitous phenomenon whose exact function remains unclear in many cases. To identify novel functions, this paper provides an exploration of general dynamical mathematical models of homo-oligomerisation. Simulation and analysis of these models show that homo-oligomerisation on its own allows for a remarkable variety of complex dynamic and steady-state regulatory behaviour such as transient overshoots or homeostatic control of monomer concentration. If post-translational modifications are considered, however, conventional mass-action kinetics leads to thermodynamic inconsistencies due to asymmetric combinatorial expansion of reaction routes. Introducing a conservation principle to balance rate equations re-establishes thermodynamic consistency. Using such balanced models it is shown that oligomerisation can lead to bistability without feedback by enabling pseudo-multisite modification and kinetic pseudo-cooperativity via multi-enzyme regulation. Thereby, oligomerisation significantly reduces the requirements previously thought necessary for the occurrence of bistability.


Introduction
Homo-oligomerisation of proteins, i.e. the assembly of supramolecular protein complexes made up from multiple identical subunits, is a ubiquitous phenomenon. In vertebrates, about 30-50% of all proteins form homo-oligomers consisting of two or more subunits [1,2]. Oligomerisation may oer several advantages: it can be a way to economically assemble larger structures (thereby reducing genome size) and allows for a higher error-free transcription chance for individual subunits. Moreover, it can provide additional regulatory control via allostery and cooperative binding events (hemoglobin being the classical example) [3,4]. Yet, in many cases the function of homooligomerisation remains unclear.
Dynamical mathematical models based on ordinary dierential equations (ODEs) have been extensively used to study many important motifs, mechanisms and phenomena in signal transduction networks. To lesser extent, ODE models have also been used to study signal transduction processes involving homo-oligomers.
Such theoretical studies have shown that in addition to the well-known role in the emergence of ultrasensitive responses via cooperative binding, oligomerisation can provide an additional layer of control over such responses. Bouhaddou an Birtwistle, for instance, showed that dierent oligomerisation routes provide an eective means of tuning ultrasensitive cooperative responses [5]. Buchler and Louis showed that homo-oligomerisation itself can lead to modest ultrasensitivity independently from cooperativity [6]. If coupled to positive feedback, the ultrasensitivity generated e.g. by homo-dimerisation is sucient for the emergence of bistability [7]. For signalling involving dimeric receptors and substrate activation, the presence of a single/dual activation mechanism can lead to complex, non-linear signal dynamics [8]. Taken together, this highlights the importance of homooligomerisation and the use of mathematical modelling as a tool to study their roles in signal transduction.
However, above mentioned studies focussed on specic questions, contexts or systems. A general analysis of homo-oligomerisation in terms of assembly dynamics, steady state behaviour and the potential eects of post-translational modications (PTMs) is neither covered by classical and popular textbooks on mathematical or system's biology (see e.g. [913]), nor is the author aware of such an analysis in the recent research literature. It thus seems that an exploration of general dynamical mathematical models of homooligomerisation and is still lacking. This paper provides such an exploration. As this study focusses solely on homo-oligomerisation, we will leave out the prex homo-' in the remainder of this article for the sake of brevity.
Starting with simple mass action kinetics based models of dimerisation to tetramerisation, we will study assembly dynamics and steady state behaviour numerically. Even though the rst presented models are very simple, it is found that they are capable of complex dynamical and steady state behaviour such as undulations and homeostatic regulation.
Next, PTMs of oligomers are considered. Surprisingly, the application of conventional mass action rate laws easily results in thermodynamically inconsistent models due to combinatorial expansion of the oligomerisation routes upon modication. The issue can be circumvented by balancing the rate expressions based on a rate conservation principle.
Finally, two novel mechanisms based on oligomerisation which lead to ultrasensitive, bistable PTM responses will be presented: pseudo-multisite modication and regulation by multiple enzymes.
The focus of the current work is to demonstrate that oligomerisation enables more complex regulatory behaviour than previously appreciated. While the broad scope of a general analysis of dynamical mathematical models of oligomerisation does not permit an exhaustive treatment of all aspects within the limit of a single article, some of the most important implications and avenues for future research will be outlined in the discussion.

Results
Simple mass action models of oligomerisation: ultrasensitivity and homeostasis Let us begin by assuming that a general protein A can form oligomers with a maximum number of n subunits Figure 1 Model scheme of homo-tetramerisation based on conventional mass action kinetics assuming that all intermediate species (dimers and trimers) are possible in the reaction pathway. See text for the dierential equations describing the system.
(protomers) per oligomeric complex. We furthermore assume that A can form all intermediate oligomeric and that each oligomeric species is formed through simple one-step, second-order binding reactions described by mass action kinetics. For the remainder of this article, we will study oligomers with a maximum of four protomers or less, i.e. tetramers, trimers and dimers.
It is likely that many of the presented ndings apply to higher-order oligomers as well.
In the case of tetramers we therefore assume that tetramers can be formed by the association of two dimers or, alternatively, of a trimer and a monomer.
The total amount of subunits is conserved by the re- and can be used to eliminate one of the above equations. Note that models for tri-or dimerisation can be obtained simply by removing reactions R5-R8 or R3-R8, respectively. For the sake of simplicity, we will begin by assuming equal rate constants of 10 7 mol −1 s −1 for all association reactions and equal rate constants of 10s −1 for all dissociation reactions, thereby yielding a changes in total protein concentration, and that ultrasensitivity can increase with higher number of protomers per complex (as can also be seen from the increasing slopes in Figure 2E,F) [6]. Steady state analysis of trimerisation and tetramerisation models with varied parameters. The relative change of parameters is visualised in the upper reaction schemes. Parameters: we will see shortly, even accounting merely for a single PTM makes model formulation of anything higher than dimers unlikely more dicult due to the combinatorial expansion of potential oligomerisation routes.
In the following, we will therefore restrict mass action models to a maximum of trimerisation.
Unfortunately, combinatorial expansion is not the only challenge when PTMs of oligomers are considered: it is remarkably easy to slip into thermodynamical inconsistency even with models based purely on conventional mass action kinetics. Before we incorporate PTMs into our models we will therefore formulate some biochemical intuitions and expectations which will later guide us to avoid thermodynamic inconsistency.
Let us suppose an oligomeric protein can be mod- .., n}, can be employed to describe the rate of consumption v i of substrate S i [16]. That is, the individual substrates act as competitive inhibitors for each other. We are now able to formulate reaction schemes, reaction rates and model equations. Figure 4A shows the reaction scheme and rate expressions for the dimerisation model based on mass action kinetics for oligomerisation and mentioned Michaelis-Menten type rate law for addition and removal of PTMs. The equations are: In other words: from the perspective of the oligomerisation equilibrium, all PTM isoforms of a n-tamer are treated as a single species. Conversely, the sum of all rates v ′ i , 1 ≤ i ≤ j, of reactions dissociating A n of any modication status is equal to the dissociation rate based on A n 's total concentration. That is, at all times We call the right-hand side of each identitiy the eective rates. and How does this principle help us to avoid thermodynamic inconsistency? The principle relates the j reaction rates from the reaction scheme (left-hand side (LHS) of (1) and (2)) to the thermodynamically expected eective rates (right-hand side (RHS) of (1) and (2)). Instead of assigning a priori rate expressions based on mass action kinetics, we will assign rates only after making sure the principle is not violated. In the rst step of this check we will expand the RHS of the rate conservation identity. In a second step, we will substitute the rates on the LHS with their mass action kinetics expression. Next, we will compare both sites: if they are equal, all rates can readily be identied with their mass action kinetics expression. If they are not equal, we balance the terms on the LHS where the discrepancy occurs by introducing coecients that ensure the validity of the identity. Lastly, we identify the respective rates with the balanced terms. Let us illustrate this using equations (1) and (2) derived from the dimerisation scheme: (1) → expanding RHS, substituting LHS: → balancing deviating terms in LHS: → assigning reaction rates: (2) → expanding RHS, substituting LHS: → no balancing necessary → assigning reaction rates: It is straightforward to apply the same procedure to the trimerisation model (cf. supplementary section 2).
Other things being equal, the dimerisation and trimerisation model updated with the balanced rate expressions exhibit neither transient changes in the oligomerisation upon modication ( Figure 5D), nor shifts in the apparent K d of equimolar mixtures of unmodied/modied A ( Figure 5E). The distribution of PTM isoforms at equilibrium is binomial ( Figure 5F). Taken together, this indicates that the balanced models are indeed thermodynamically consistent.
The postulated rate conservation principle is no fundamental law and can readily be proven using the principle of detailed balance [17] as lemma (cf. supplementary section 3).
For a more intuitive understanding of this balancing procedure, it might be helpful to appreciate its similarity to stoichiometric balancing. Consider, for example, the chemical equation for the reaction of oxygen with hydrogen to water: As the reader will have spotted, there are two oxygen atoms on the LHS, but only one on the RHS of the equation. As this violates the law of conservation of mass, we need to balance the equation by adding stoichiometric coecients: The rate balancing procedure presented in this paper is conceptually identical: As the PTM is assumed not to inuence oligomerisation, we know that total formation and dissociation rates of oligomers are conserved; they must be the same as for unmodied protein. However, the PTM increases the number of possible oligomeric species due to combinatorial expansion. This expansion is asymmetric because there are more possibilities to combine modied and unmodied subunits to a n-tamer the higher its order: two for a monomer, three for a dimer, four for a trimer and so forth (cf. Figure 4B). This creates additional oligomerisation routes and alters the net-rates of oligomer formation and dissociation, thereby violating the rate conservation principle. To avoid this, we need to balance this purely combinatorial eect by introducing balancing coecients for the rate expressions.
Considering PTMs: ultrasensitivity and bistability Now that we have thermodynamically consistent models of oligomers which can be post-translationally (de-)modied, we will explore the steady state behaviour in the presence of (de-)modication enzymes E1, E2 using the dimer model as an example. The relative fraction of modied dimer and monomer shows pronounced ultrasensitivity in response to increasing concentrations of modifying enzyme E1 ( Figure 6A).
On closer examination, this is not very surprising.
Apart from some degree of zero-order ultrasensitivity arising from enzyme saturation [18], oligomerisation additionally creates a substrate competition situation between monomeric and oligomeric species for tion steps is smaller than the product of the rate constants for the second modication and demodication steps [24]. Without considering the oligomeric nature, introducing positive kinetic cooperativity for the demodication of the dimer, i.e. assuming k 2 > k 4 , would fulll this requirement. Indeed, implementing this assumption leads to bistability with respect to the modication status in the dimer model ( Figure 6B). As the bistable range increases with the number of cooperative modication steps [22], the likelihood for a bistable PTM status will increase with higher order oligomers.
Interestingly, not only the dimer, but also the monomer exhibits bistability even without multiple sites for PTMs. This becomes less surprising if one considers that the dimer is in equilibrium with the monomer allowing modied dimers to dissociate into monomers. Furthermore, when dimers are completely (de-)modied, substrate competition for (de-) modication of the monomer abates, allowing for more monomer (de-)modication.
While perhaps not uncommon, kinetic cooperativity might not be the only way to realise bistability in instance, assume that in a dually modied dimer, each PTM mutually prevents (e.g. due to sterical reasons) access to the other PTM for demodifying enzyme E3.
Only when one of the PTMs has already been removed by demodifying enzyme E2 (which we assume to catalyse PTM removal from the singly and dually modied dimer equally well), can E3 bind to the singly modied dimer and catalyse the last demodication step.
Assuming that E3 can also catalyse demodication of the modied monomer, the scheme for updated dimer model is shown in Figure 7A. for dierent oligomeric species [26]. As it is also involved in dynamic signal encoding leading to dierent cell-fate decisions [27], it is tempting to speculate that some of this could be the result of oligomerisation. Another promising candidate for dynamic signal encoding through oligomerisation could be the EGFreceptor for which dimers, trimers and tetramers have been described [28,29]. A considerable list of higherorder homo-oligomers for which various intermediate forms have been observed (and thereby might also be candidates for dynamic signal encoding) can be found in [30].
In addition to the previously described but mod- Both types of experiments can be technically challenging and likely need to be analysed through model tting [33,34].

Combinatorial complexity
As the order of oligomers increases and/or PTMs are taken into account, the number of species and possible reactions quickly grows. This is a typical situation of`combinatorial explosion' which poses a signicant challenge for many signal transduction models [35,36].  [19,21,22,24], this possibility seems obvious from a biochemical point of view, yet, has not been appreciated before. An interesting and unique twist of this motif is that the bistability resulting from modication of the oligomer extends to the monomer due to intrinsic substrate competition and because both species are in equilibrium with each other. We also demonstrated that kinetic cooperativity of multisite modication systems is not a requirement for bistability. If multiple enzymes regulate the modication steps and if some can only catalyse a subset of the individual modication steps, this leads eectively to the same kinetic asymmetry [22,24] as kinetic cooperativity. While oligomers might be particularly suited for this mechanism due their symmetrical structure, bistability through multienzyme regulation could in principle arise in any multisite PTM system.
The relevance of these ndings is that they sig-

Conclusion
We have demonstrated that homo-oligomers, making up approximately 30-50% of the proteome [1,2], oer an even greater variety of regulatory mechanisms than previously appreciated. It is likely that some of these are relevant to many cellular signalling pathways. It also may partly explain why homo-oligomerisation is so commonly found throughout evolution. Hopefully, the presented ndings will be helpful to modellers interested in homo-oligomeric signalling proteins and stimulate experimental research into signalling processes to which the presented ndings might apply.

Methods
Details on the computational procedures can be found in supplementary section 4.