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

bioRxiv preprint doi: https://doi.org/10.1101/758789; this version posted October 24, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC 4.0 International license. Homo-Oligomers in Signal Transduction: Dynamics, Homeostasis, Ultrasensitivity, Bistability

Daniel Koch orcid.org/0000-0002-4893-1774 E-mail: [email protected]

London, October 2019

Abstract: Homo-oligomerisation of proteins is a ubiquitous phenomenon. Previous research showed that homo-dimerisation can generate ultrasensitivity which can lead to bistability when coupled to positive feedback. This suggests that homo-oligomerisation can play important roles in signal transduction beyond its well-known role in cooperative binding. To identify novel functions, this paper provides an exploration of general dynamical mathematical models of homo-oligomerisation. Simulations and model analyses 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 modications are considered, conventional mass-action kinetics leads to thermodynamic inconsistencies due to combinatorial expansion of reaction routes. Introducing a conservation principle to balance rate equations re-establishes thermodynamic consistency. Moreover, it is shown that oligomerisation can lead to bistability without feedback by enabling pseudo-multisite modication and kinetic pseudo-cooperativity via multi-enzyme regulation. Thereby, oligomerisation signicantly reduces the requirements previously thought necessary for the occurrence of bistability.

Contents

Introduction 2

Results 2 Simple mass action models of oligomerisation: ultrasensitivity and homeostasis...... 2 Considering PTMs: ensuring thermodynamical consistency ...... 4 Considering PTMs: ultrasensitivity and bistability ...... 10

Discussion 11

Methods 13

References 14

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Daniel Koch Homo-Oligomers in Signal Transduction

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

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Daniel Koch Homo-Oligomers in Signal Transduction

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 Kd value of 1 µM for all reactions, a typical value for assume that A can form all intermediate oligomeric many protein-protein interactions. species with m subunits (where m ∈ , 1 < m < n) N Time course simulations of the system with initial and that each oligomeric species is formed through sim- conditions [A] = A = 10µM show association dy- ple one-step, second-order binding reactions described 0 tot namics typical for binary protein-protein interactions by mass action kinetics. For the remainder of this ar- in the case of dimerisation, whereas trimerisation and ticle, we will study oligomers with a maximum of four tetramerisation reactions exhibit a transient overshoot protomers or less, i.e. tetramers, trimers and dimers. of dimers followed by a slower decrease and increase It is likely that many of the presented ndings apply in trimers and tetramers, respectively (Figure 2A-C). to higher-order oligomers as well. The amplitude and position of such overshoots depend In the case of tetramers we therefore assume that strongly on the monomer concentration at the begin- tetramers can be formed by the association of two ning of the reaction (Figure 2C, inset). More complex dimers or, alternatively, of a trimer and a monomer. dynamics such as dampened oscillations/undulations The reaction scheme and the reaction rates v for the i on dierent timescales are possible (Figure 2B, inset). individual reactions can then be summarised as in Fig- If the individual oligomeric species possess dierent bi- ure 1. Denoting the monomeric to tetrameric species ological functionality, such dynamics could be a mech- by A, ..., AAAA we can now formulate the system's anism for dynamic signal encoding as will be outlined equations: in the discussion in more detail. d [A] = 2 ⋅ v2 + v4 + v6 − 2 ⋅ v1 − v3 − v5 dt Numerical steady state analysis shows that the d [AA] = v1 + v4 + 2 ⋅ v8 − v2 − v3 − 2 ⋅ v7 dt dose-response curves for the individual species meet d [AAA] = v3 + v6 − v4 − v5 dt in a single intersection point (Figure 2D-F), mirroring d [ ] = + − − dt AAAA v5 v7 v6 v8 the assumption that all reactions have the same Kd

The total amount of subunits is conserved by the re- value. Local sensitivity analysis at Atot = 0.1µM with dS Atot lation Atot = [A] + 2 ⋅ [AA] + 3 ⋅ [AAA] + 4 ⋅ [AAAA] 2% perturbation yields relative sensitivities dAtot S and can be used to eliminate one of the above equa- (where S is the steady state concentration of ei- tions. Note that models for tri- or dimerisation can ther A, ..., AAAA) of 0.87 for monomers and 1.76 for be obtained simply by removing reactions R5-R8 or dimers in the dimerisation model, 2.58 for trimers in R3-R8, respectively. For the sake of simplicity, we will the trimersation model and 3.46 for tetramers in the begin by assuming equal rate constants of 107mol−1s−1 tetramerisation model. The analysis conrms again for all association reactions and equal rate constants of that oligomerisation at concentrations below the Kd 10s−1 for all dissociation reactions, thereby yielding a can lead to modest ultrasensitivity in response to

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Daniel Koch Homo-Oligomers in Signal Transduction

Figure 2 Time course simulations (A-C) and steady state analysis (D-F) for dimers (A,D), trimers (B,E) and tetramers (C,F). Initial con- 7 −1 −1 −1 ditions for A-C: [A]0 = 10µM, [AA]0 = [AAA]0 = [AAAA]0 = 0M. Parameters: (A and D) k1 = 10 mol s , k2 = 10s , (B 7 −1 −1 −1 7 −1 −1 9 −1 −1 and E) k1 = k3 = 10 mol s , k2 = k4 = 10s , (C and F) k1 = k3 = k5 = k7 = 10 mol s (C inset: k3 = 10 mol s ), −1 k2 = k4 = k6 = k8 = 10s . changes in total protein concentration, and that ultra- the coin of oligomeric ultrasensitivity. If we consider sensitivity can increase with higher number of pro- the monomer concentration at higher total protein con- tomers per complex (as can also be seen from the in- centrations in Figure 3C, it becomes apparent that creasing slopes in Figure 2E,F) [6]. oligomerisation can be an ecient homeostatic regu- For many proteins able to form higher order latory mechanism of the monomer concentration (rela- oligomers, the presence of a single or a small subfrac- tive sensitivity of 0.2 for monomers at Atot = 100µM). tion of possible oligomeric species often dominates over This would be plausible in situations where monomers other potential intermediate species, indicating that are the biologically active species. Note that this mechanism does not require a complex feedback organisa- oligomerisation is often cooperative and that Kd values dier for the individual reactions. Varying the model tion typically associated with homeostasis [14,15]. parameters in a way to favour formation of the highest order oligomer in the trimerisation and tetramerisa- Considering PTMs: ensuring thermodynamical tion model (e.g. by increasing association rate con- consistency stants) can reproduce the dominance of the highest order oligomer over large concentration ranges (Fig- Just like non-oligomeric proteins, oligomeric proteins ure 3A,B). This also leads to a shift of intersection are subject to various post-translational modications points, resulting in dierent apparent Kd values be- such as phosphorylation, ubquitinylation, lipidation tween the individual intermediate oligomerisation re- and others. Sometimes these modications can reg- actions. Tweaking of the parameters allows to shift ulate the equilibria between monomeric and oligomeric the curves for each individual species into almost any species via conformational changes or sterical hin- direction (data not shown). drance. Other times these modications are irrele- Parameter variation also highlights the ipside of vant to the protein's oligomerisation behaviour. As

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Daniel Koch Homo-Oligomers in Signal Transduction

Figure 3 Steady state analysis of trimerisation and tetramerisation models with varied parameters. The relative change of parameters 7 −1 −1 9 −1 −1 −1 is visualised in the upper reaction schemes. Parameters: (A) k1 = 10 mol s , k3 = 10 mol s , k2 = k4 = 10s , (B) 6 −1 −1 −1 7 −1 −1 −1 9 −1 −1 −1 8 −1 −1 −8 −1 k1 = 10 mol s , k2 = 5s , k3 = 10 mol s , k4 = 50s , k5 = 10 mol s , k6 = 0.1s , k7 = 10 mol s , k8 = 5 × 10 s , 7 −1 −1 3 −1 3 −1 −1 −1 6 −1 −1 −1 (C) k1 = 9.74 × 10 mol s , k2 = 2.78 × 10 s k3 = 2.31 × 10 mol s , k4 = 0.0155s , k5 = 1.45 × 10 mol s , k6 = 0.0262s , 6 −1 −1 −6 −1 k7 = 8.73 × 10 mol s , k8 = 1.0584 × 10 s . we will see shortly, even accounting merely for a sin- dissociation constants (a3) are identical for the gle PTM makes model formulation of anything higher following oligomerisation reactions: than dimers unlikely more dicult due to the combi- only unmodied protein natorial expansion of potential oligomerisation routes. In the following, we will therefore restrict mass action only modied protein models to a maximum of trimerisation. modied with unmodied protein Unfortunately, combinatorial expansion is not the only challenge when PTMs of oligomers are consid- (b) If the modifying enzyme is added to an ered: it is remarkably easy to slip into thermodynam- equilibrated mixture of completely unmodied ical inconsistency even with models based purely on monomers and oligomers, both monomers and conventional mass action kinetics. Before we incor- oligomers will be modied over time, yet the to- porate PTMs into our models we will therefore for- tal concentrations of monomers and oligomers re- mulate some biochemical intuitions and expectations mains constant. which will later guide us to avoid thermodynamic in- (c) There are 2+n−1 possibilities to combine un- n consistency. modied/modied monomers into symmetric n- Let us suppose an oligomeric protein can be mod- tamers. Thus, if equal parts of unmodi- ied by a PTM at a single site. For the sake of sim- ed/modied monomers are mixed to form n- plicity we assume that the site lies remote from the tamers, the isoform distribution of n-tamers with oligomerisation interface and does not alter any of the 0 ≤ i ≤ n modied subunits will be binomal. reaction parameters. Intuitively, we would then expect the following to be true: To proceed with model formulation, let A∗, AA∗, AA∗∗, ... denote modied monomers, dimers (a) The association rate constants (a1), dissociation with one and dimers with two modied protomers and rate constants (a2) and therefore equilibrium and so forth. Keeping the assumption that each oligomeric

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Daniel Koch Homo-Oligomers in Signal Transduction

Figure 4 Reaction schemes for the mass action kinetics models of dimerisation (A) and trimerisation (B) including reversible post-translational modications. See supplementary information for reaction rates and ODEs of the trimersation model. species is formed through binary association reactions, where J = {1, ..., n}, can be employed to describe the it follows from (a1)-(c1) that each pair of molecular rate of consumption vi of substrate Si [16]. That is, species X,Y which is able to associate in the absence the individual substrates act as competitive inhibitors of any PTMs is also able to associate with identi- for each other. We are now able to formulate reaction cal reaction parameters regardless of how many pro- schemes, reaction rates and model equations. tomers of X or Y are modied. We will assume that Figure 4A shows the reaction scheme and rate ex- there is a modifying enzyme E1 and a demodifying pressions for the dimerisation model based on mass enzyme E2 which operate by a non-cooperative, ir- action kinetics for oligomerisation and mentioned reversible and distributive mechanism and that all Michaelis-Menten type rate law for addition and re- molecular species, regardless the number of their pro- moval of PTMs. The equations are: tomers, are (de-)modied with the same kinetic param- d [A] = 2 ⋅ v6 + v8 + v12 − 2 ⋅ v5 − v7 − v11 eters, i.e. the oligomeric state does not inuence the dt d [A∗] = v8 + 2 ⋅ v10 + v11 − v7 − 2 ⋅ v9 − v12 (de-)modication reactions. These assumptions reect dt d [AA] = v2 + v5 − v1 − v6 the situation where a PTM does not induce conforma- dt d [AA∗] = v1 + v4 + v7 − v2 − v3 − v8 tional changes and lies remote from the oligomerisation dt d [AA∗∗] = v3 + v9 − v4 − v10 interface, allowing the enzymes to access the PTM site dt equally in all oligomeric species. We therefore expect Figure 4B shows the reaction scheme for the trimeri- the individual monomeric and oligomeric species to sation model including PTMs. See supplementary sec- compete for enzymes E1 and E2. In situations with tion 1 for reaction rates and model equations. Note that the reaction scheme for trimerisation and a single multiple competing substrates S1, ..., Sn an irreversible Michaelis-Menten type rate law of the form: PTM is already more complex than the tetramerisation scheme without PTMs. For dimerisation, we will VmaxSi vi = , use the same parameter values as in Figure 2A. For ( + Sj ) + Kmi 1 ∑j∈J∖{i} K Si mj trimerisation, we will use the same parameter values as

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Daniel Koch Homo-Oligomers in Signal Transduction in Figure 3A. We chose catalytic rate constants to be and a uniform distribution of modied oligomers (Fig- −1 kcat = 1s and Michaelis constants to be Km = 1µM ure 5C). Taken together, this suggests that the tran- for all (de-)modication reactions in both models. sient changes are indeed a modelling artefact resulting Let us rst consider the eects of adding catalytic from thermodynamic inconsistency. concentrations of E1 to an equilibrated mixture of Where did we go astray if using simple mass ac- monomers and oligomers. Interestingly, as the modi- tion kinetics, arguably the most established determinis- cation reaction proceeds, transient changes in the total tic modelling approach for chemical reactions, leads to concentrations of monomeric and oligomeric species oc- thermodynamic inconsistency? The answer is: naively cur (Figure 5A; the total concentrations being the sum deriving ordinary dierential equations by applying of concentrations of all modied isoforms for a given mass action kinetics and stoichiometrical balancing to monomeric or oligomeric species). The amplitude and reaction schemes such as in Figure 4. In order to cir- direction of these transients appear to be both inu- cumvent this issue, we need to introduce a more general enced by the maximum number of protomers. Intu- notation for oligomeric species and formulate a further itively, the phenomenon results from the accessibility expectation. of additional oligomerisation routes during the modi- Let m denote a -times modied n-tamer, i.e. an An m cation reaction. Consider for example the dimerisa- 1 oligomeric complex with n ∈ N≥ identical subunits of tion reaction scheme (Figure 4A). If A is either com- ≥1 which 0 ≤ m ≤ n carry a PTM at site x. Let j ∈ N be pletely unmodied or modied, a single reaction route the number of reversible bimolecular association reac- between a monomeric and a dimeric species is avail- tions between lower-order complexes p and q able Ar As able. If a mixture of A and A∗ is present, a third to form m, where . An r + s = n route appears: the reaction between A and A∗ to Next, let I(An) = {{r1, s1}, {r2, s2}, ...} dene the set ∗ AA . We would therefore expect that decreasing the of actually occurring combinations of oligomeric or- velocity of oligomerisation/dissociation or increasing p q ders of An's educts A and A , where p + q ≤ n and the speed of modication would diminish the eect, ri si ri +si = n. For example, if a hexamer can be assembled as both changes would limit A's capacity to change its from dimers with tetramers and from monomers with oligomerisation state during the transition time from pentamers (of any PTM status), I(A6) would be the completely unmodied protein to complete modica- set {{2, 4}, {1, 5}}. tion. Indeed, while decreasing the velocity of oligomeri- From (a1) and (a2) it follows that all forward steps sation/dissociation reduces the amplitude, increasing between Ap and Aq producing Am have the same asso- the enzyme concentration limits the peak width of the ri si n ciation rate constant k and each corresponding reverse amplitude (supplementary Figure S1). i step has the same dissociation rate constant k′ for all Clearly, these transient changes contradict expec- i possible values of m, p and q. Unaltered rate constants tation (b) and no experimental data so far (as far as imply unaltered equilibrium constants. Therefore, we the author is aware) reported transient alterations of expect that the oligomerisation state upon post-translation without aecting association/dissociation rates. To fur- (d) the PTM status is irrelevant for the equilibrium. ther explore the models and to test whether the tran- Thus, the equilibrium is solely determined by the sient changes could be a modelling artefact, let us see total concentrations of each n-tamer, i.e. the sum what happens if equimolar mixtures of modied and of all PTM-isoforms of An: unmodied are simulated to steady state. Accord- A n [ ] = [ m] ing to (a3) we would expect Kd values to be identical An,tot Q An to that of completely unmodied and completely mod- m=0 ied A. We would furthermore expect a binomial dis- In other words: from the perspective of the oligomeri- tribution of modied oligomers according to (c). How- sation equilibrium, all PTM isoforms of a n-tamer are ever, both models show altered Kd values (Figure 5B) treated as a single species.

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Daniel Koch Homo-Oligomers in Signal Transduction

Figure 5 Behaviour of di- and trimerisation models upon post-translational modication. (A,D) timecourses of total monomer and total oligomer concentrations after addition of 50nM [E1] starting with [Atot] = 10µM at equilibrium. Black-dotted curves show progress of the modication reaction. (B,E) apparent dissociation constants and (C,F) distribution of PTM isoforms of the highest order oligomer in equimolar mixtures of modied and unmodied protein at equilibrium.

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Daniel Koch Homo-Oligomers in Signal Transduction

For a graphical illustration of the denitions and the rate conservation identity. In a second step, we will equilibrium situation see supplementary Figure S2. We substitute the rates on the LHS with their mass ac- can now formulate the following principle: tion kinetics expression. Next, we will compare both sites: if they are equal, all rates can readily be identi- Conservation of oligomerisation rates ed with their mass action kinetics expression. If they If a PTM does not inuence the parameters of are not equal, we balance the terms on the LHS where an oligomerisation reaction, the sum of all rates the discrepancy occurs by introducing coecients that ensure the validity of the identity. Lastly, we identify vi, 1 ≤ i ≤ j, of reactions leading to a n-tamer An of any modication status is equal to the asso- the respective rates with the balanced terms. Let us ciation rates based on the total concentrations illustrate this using equations (1) and (2) derived from the dimerisation scheme: of An's educts (i.e. all modication isoforms). Conversely, the sum of all rates ′, , vi 1 ≤ i ≤ j (1) → expanding RHS, substituting LHS: of reactions dissociating An of any modication 2 ∗ ∗ 2 status is equal to the dissociation rate based on k1[A] + k1[A][A ] + k1[A ] = [ ]2 + [ ][ ∗] + [ ∗]2 An's total concentration. That is, at all times k1 A 2k1 A A k1 A

j r s v = (k ⋅ [Am] ⋅ [Am]), → balancing deviating terms in LHS: Q i Q i Q ri Q si i=1 i∈I m=0 m=0 2 ∗ ∗ 2 k1[A] + 2 ⋅ k1[A][A ] + k1[A ] and 2 ∗ ∗ 2 = k1[A] + 2k1[A][A ] + k1[A ] j n ′ ′ m Q vi = Q(ki ⋅ Q [An ]). i=1 i∈I m=0 → assigning reaction rates:

We call the right-hand site of each identitiy the 2 ∗ v5 = k1[A] , v7 = 2k1[A][A ] eective rates. ∗ 2 v9 = k1[A ] The principle's name is chosen in analogy to the conservation of mass as it conserves the reaction rates at (2) → expanding RHS, substituting LHS: given total concentrations of educts regardless the dis- ′ ∗ ′ ∗∗ k1[AA] + k1[AA ] + k1[AA ] tribution of PTM isoforms. Although it might appear = k′ [AA] + k′ [AA∗] + k′ [AA∗∗] complicated, it is straightforward to illustrate the prin- 1 1 1 ciple using dimerisation as an example. Applying it to → no balancing necessary Figure 4A yields and thus I(A2) = {{1, 1}} → assigning reaction rates:

v5 + v7 + v9 = k ([A] + [A∗])2 (1) ′ ′ ∗ 1 v6 = k1[AA], v8 = k1[AA ] v10 = k′ [AA∗∗] and v6 + v8 + v10 = 1 ′ ∗ ∗∗ (2) It is straightforward to apply the same procedure to k1([AA] + [AA ] + [AA ]). the trimerisation model (cf. supplementary section 2). How does this principle help us to avoid thermo- Other things being equal, the dimerisation and trimeri- dynamic inconsistency? The principle relates the j re- sation model updated with the balanced rate expres- action rates from the reaction scheme (left-hand side sions exhibit neither transient changes in the oligomeri- (LHS) of (1) and (2)) to the thermodynamically ex- sation upon modication (Figure 5D), nor shifts in pected eective rates (right-hand side (RHS) of (1) the apparent Kd of equimolar mixtures of unmodi- and (2)). Instead of assigning a priori rate expressions ed/modied A (Figure 5E). The distribution of PTM based on mass action kinetics, we will assign rates only isoforms at equilibrium is binomial (Figure 5F). Taken after making sure the principle is not violated. In the together, this indicates that the balanced models are rst step of this check we will expand the RHS of the indeed thermodynamically consistent.

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Daniel Koch Homo-Oligomers in Signal Transduction

The postulated rate conservation principle is no On closer examination, this is not very surprising. fundamental law and can readily be proven using the Apart from some degree of zero-order ultrasensitivity principle of detailed balance [19] as lemma (cf. supple- arising from enzyme saturation [17], oligomerisation mentary section 3). additionally creates a substrate competition situa- For a more intuitive understanding of this balancing tion between monomeric and oligomeric species for procedure, it might be helpful to appreciate its similar- (de-)modication and provides pseudo-multisites for ity to stoichiometric balancing. Consider, for example, PTMs (i.e. multiple protomers with identical PTM the chemical equation for the reaction of oxygen with sites). Both motifs are capable of generating ultra- hydrogen to water: sensitivity [18, 20]. Moreover, multisite modication can in principle generate bistability if there is a su- O2 + H2 → H2O cient asymmetry in the sequential modication cycles, As the reader will have spotted, there are two oxy- i.e. if either the demodication and/or the modica- gen atoms on the LHS, but only one on the RHS of tion steps exhibit kinetic cooperativity [21, 22]. For the equation. As this violates the law of conservation dual-site modication of monomeric proteins, Conradi of mass, we need to balance the equation by adding and Mincheva have proven that in general, bistability stoichiometric coecients: must occur for some concentrations of demodifying and

O2 + 2H2 → 2H2O modifying enzyme if the product of the rate constants for the rst modication and demodication steps is The rate balancing procedure presented in this pa- smaller than the product of the rate constants for the per is conceptually identical: As the PTM is assumed second modication and demodication steps. With- not to inuence oligomerisation, we know that total out considering the oligomeric nature, introducing pos- formation and dissociation rates of oligomers are con- itive kinetic cooperativity for the demodication of the served; they must be the same as for unmodied pro- dimer, i.e. assuming , would fulll this require- tein. However, the PTM increases the number of pos- k2 > k4 ment. Indeed, implementing this assumption leads to sible oligomeric species due to combinatorial expan- bistability with respect to the modication status in sion. This expansion is asymmetric because there are the dimer model (Figure 6B). As the bistable range more possibilities to combine modied and unmodied increases with the number of cooperative modication subunits to a n-tamer the higher its order: two for a steps [22], the likelihood for a bistable PTM status will monomer, three for a dimer, four for a trimer and so increase with higher order oligomers. forth (cf. Figure 4B). This creates additional oligomerisation routes and alters the net-rates of oligomer for- Interestingly, not only the dimer, but also the mation and dissociation, thereby violating the rate con- monomer exhibits bistability even without multiple servation principle. To avoid this, we need to balance sites for PTMs. This becomes less surprising if one this purely combinatorial eect by introducing balanc- considers that the dimer is in equilibrium with the ing coecients for the rate expressions. monomer allowing modied dimers to dissociate into monomers. Furthermore, when dimers are completely Considering PTMs: ultrasensitivity and bista- (de-)modied, substrate competition for (de-) mod- bility ication of the monomer abates, allowing for more Now that we have thermodynamically consistent mod- monomer (de-)modication. els of oligomers which can be post-translationally While perhaps not uncommon, kinetic cooperativ- (de-)modied, we will explore the steady state be- ity might not be the only way to realise bistability in haviour in the presence of (de-)modication enzymes (pseudo-)multisite PTM systems. From a biochemical E1, E2 using the dimer model as an example. The rel- point of view, asymmetry in the (de-)modication rate ative fraction of modied dimer and monomer shows of a multi-site PTM protein could eectively be re- pronounced ultrasensitivity in response to increasing alised, too, if one of the (de-)modication steps would concentrations of modifying enzyme E1 (Figure 6A). also be catalysed by another enzyme E3. Let us, for

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Daniel Koch Homo-Oligomers in Signal Transduction

Figure 6 Ultrasensitivity and Bistability of modication response. (A) fractional modication of both monomers and dimers in response to increasing concentrations of modifying enzyme E1 is notably ultrasensitive. (B) left, time course simulations demonstrate that the approached steady state is determined by the initial conditions if demodication is assumed to be cooperative. Parameters: −1 −1 k2 = 100s , k4 = 10s , [E1] = 3µM, [E2] = 0.1µM, Atot = 10µM with dierent fractional modication at t = 0), other parameters as specied for Figure 2A. (B) middle and right, bifurcation diagrams show identical parameter values for saddle node bifurcations of dimer and monomer modication. Unstable steady states are indicated by dotted lines. instance, assume that in a dually modied dimer, each tutive monomer/oligomer equilibrium. Examples in- PTM mutually prevents (e.g. due to sterical reasons) clude membrane receptors which oligomerise upon lig- access to the other PTM for demodifying enzyme E3. and binding [24] or proteins that oligomerise upon re- Only when one of the PTMs has already been removed cruitment to a membrane. If dimers and higher order by demodifying enzyme E2 (which we assume to catal- oligomers have dierent downstream signalling func- yse PTM removal from the singly and dually modi- tions, such transients could be an eective way to en- ed dimer equally well), can E3 bind to the singly code the duration of the input signal (e.g. ligand pres- modied substrate and catalyse the last demodica- ence or membrane recruitment) and thereby lead to dif- tion step. Assuming that E3 can catalyse demodica- ferent cellular responses for short and prolonged stim- tion of the modied monomer, the scheme for updated uli. The tumor suppressor p53 is a relevant example dimer model is shown in Figure 7A. Using the updated of a protein dierent biological activity for dierent dimer model, it is not dicult to nd parameter val- oligomeric species [25]. As it is also involved in dy- ues that lead to bistability (Figure 7B), showing that namic signal encoding leading to dierent cell-fate de- multi-enzyme regulation can be an eective alternative cisions [26], it is tempting to speculate that some of for realising the asymmetry required for bistability in this could be the result of oligomerisation. multisite PTM systems. In addition to the previously described but modest ultrasensitivity through oligomerisation [6, 7], we Discussion have seen that oligomerisation could also be an eective Complex dynamics and steady state behaviour homeostatic regulatory mechanism to keep monomer Even simple oligomerisation systems can in principle be concentration in a narrow range. In contrast to the dy- capable of surprisingly complex behaviour. Dynamical namical phenomena, this more likely applies to proteins phenomena such as transient overshoots of dimers fol- which are in a constitutive monomer/oligomer equilib- lowed by a slower increase in higher order oligomers rium. Recently, Frieden proposed oligomerisation as will be relevant to proteins which are not in consti- metabolic control mechanism [27]. Given that many

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Daniel Koch Homo-Oligomers in Signal Transduction

Figure 7 Bistability through multi-enzyme regulation of modication status of oligomers. (A) scheme of (balanced) dimer model with additional demodication enzyme E3 which can not catalyse the rst step of the dimer demodication. (B) time course simulations −1 and bifurcation plots demonstrating bistability in the dimer model. Parameters: [E2] = 10nM, [E3] = 0.1µM, k13 = 0.1s , k14 = −1 100s ,K13 = 10µM, , K14 = 1µM, other parameters as specied for Figure 2A. enzymes oligomerise, monomer-homeostasis could be a studied experimentally enough extensively to exclude good example. If enzyme function is inhibited e.g. by scenarios such as depicted in Figure 3C. Since individ- active site obstruction in an oligomeric complex [28], ual species concentrations in the homeostatic scenario monomer-homeostasis could ensure a nearly constant often dier by ≥ 2 orders of magnitude, experimental performance of a metabolic activity over a wide range testing of such behaviour would at least require to of total protein concentration (and therefore cellular determine the equilibrium distribution of monomeric conditions such as starvation or dierent cell cycle and oligomeric species over several orders of magni- phases). tude of total protein concentration but ideally also For many tetrameric proteins, however, dimers kinetic data on oligomer (dis-)assembly which can be are the only observed intermediate species between technically challenging [32, 33]. Secondly, the possibil- monomers and tetramers. Powers and Powers used ity of monomer homeostasis via oligomerisation is not a mathematical model similar to the presented mass restricted to tetramers and could be realised by any action tetramerisation model here to explain this ob- higher order oligomersation, provided enough reaction servation [29]. Their analysis suggested that the routes allow to buer changes in the monomer con- tetramerisation pathway via trimers might `trap' many centration. A considerable list of higher-order homo- molecules in intermediate forms leading to incomplete oligomers for which various intermediate forms have tetramerisation, which, so the authors argue, might been observed can be found in [34]. An interesting lead to unfavourable consequences such as proteolysis, question for future research is whether homeostatic misfolding or aggregation. regulation could also apply to dimers or trimers if there are enough higher order oligomeric species. At rst, this may speak against oligomerisation as a homeostatic mechanism at least for tetramers. How- ever, more recent studies provided evidence that some Combinatorial complexity proteins such as the EGF-receptor can form dimers, As the order of oligomers increases and/or PTMs are tetramers and trimers, too [30, 31]. In addition, few taken into account, the number of species and possi- oligomeric proteins, including tetramers, have been ble reactions quickly grows. This is a typical situation

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Daniel Koch Homo-Oligomers in Signal Transduction of `combinatorial explosion' which poses a signicant steps, this leads eectively to the same kinetic asym- challenge for many signal transduction models [35,36]. metry [22,23] as kinetic cooperativity. While oligomers If PTMs are not considered and only one oligomeric might be particularly suited for this mechanism due species is relevant, oligomerisation pathways can be their symmetrical structure, bistability through multi- approximated via generalised mass action rate laws enzyme regulation could in principle arise in any mul- (i.e. power-laws) if the range total concentrations suf- tisite PTM system. ciently restricted (data not shown). The relevance of these ndings is that they sig- Upon inclusion of PTMs, the combinatorial expan- nicantly expand the range of contexts in which one sion of possible oligomerisation routes posed another should look for biochemical `switches' as both homo- unanticipated challenge: ensuring thermodynamic con- oligomerisation and multi-enzyme regulation are ex- sistency of the model. Rate balancing oers a solution tremely common. Phosphatases and GTPase activat- which is straightforward to apply to mass action kinet- ing proteins (GAPs), just to name two examples, are ics models. An open question is how this procedure known to often act promiscuously on multiple sub- fares if PTMs do aect oligomerisation parameters. A strates [3840]. As a consequence many phosphory- plausible conjecture would be that once the balancing lation sites can be dephosphosphorylated by multi- coecients have been introduced into the rate equa- ple phosphatases and many small GTPases (some of tions, changing parameter values for individual reac- which can also dimerise), can be inactivated by multi- tions will not aect thermodynamic consistency. ple GAPs, creating potential situations in which bista- For practical purposes, modelling higher-order bility could occur. Alternatively, multi-enzyme regu- oligomers with multiple PTM sites will generally re- lation of the modication rather than demodication quire implicit modelling approaches. Rule based mod- steps is also conceivable. Phosphorylating a protomer elling, for instance, has been applied successfully for within a dimer, for example, could lead to a new bind- modelling EGF-receptor oligomerisation [37]. ing site for a second kinase facilitating phosphorylation of the same residue in the other protomer. The combi- Bistability nation of both mechanisms, oligomerisation and multi- Ultrasensitivity and bistability are important proper- enzyme regulation, therefore represents an interesting ties in signal transduction networks for cellular deci- novel signalling motif that does not require feedback or sion making, allowing to respond in a switch-like, bi- kinetic cooperativity to generate bistable responses. nary and sometimes irreversible fashion. Oligomerisa- Conclusion tion can also lead to ultrasensitivity and bistability by We have demonstrated that homo-oligomers, making providing pseudo-multisite complexes (i.e. complexes up approximately 30-50% of the proteome [1, 2], of- with multiple identical PTM sites). Given previous fer an even greater variety of regulatory mechanisms work on ultrasensitivity and bistability arising through than previously appreciated. It is likely that some multisite modication from the Kholodenko Lab and of these are relevant to many cellular signalling path- others [18, 2123], this possibility seems obvious from ways. Hopefully, the presented ndings will be helpful a biochemical point of view, yet, has not been appre- to modellers interested in homo-oligomeric signalling ciated before. An interesting and unique twist of this proteins and stimulate experimental research into sig- motif is that the bistability resulting from modication nalling processes to which the presented ndings might of the oligomer extends to the monomer due to intrin- apply. sic substrate competition and because both species are in equilibrium with each other. We also demonstrated Methods that kinetic cooperativity of multisite modication systems is not a requirement for bistability. If multiple en- Details on the computational procedures can be found zymes regulate the modication steps and if some can in supplementary section 4. only catalyse a subset of the individual modication

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Daniel Koch Homo-Oligomers in Signal Transduction

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