Unlimited multistability in multisite phosphorylation systems

Abstract

Reversible phosphorylation on serine, threonine and tyrosine is the most widely studied posttranslational modification of proteins. The number of phosphorylated sites on a protein (n) shows a significant increase from prokaryotes, with n </= 7 sites, to eukaryotes, with examples having n >/= 150 sites. Multisite phosphorylation has many roles and site conservation indicates that increasing numbers of sites cannot be due merely to promiscuous phosphorylation. A substrate with n sites has an exponential number (2(n)) of phospho-forms and individual phospho-forms may have distinct biological effects. The distribution of these phospho-forms and how this distribution is regulated have remained unknown. Here we show that, when kinase and phosphatase act in opposition on a multisite substrate, the system can exhibit distinct stable phospho-form distributions at steady state and that the maximum number of such distributions increases with n. Whereas some stable distributions are focused on a single phospho-form, others are more diffuse, giving the phospho-proteome the potential to behave as a fluid regulatory network able to encode information and flexibly respond to varying demands. Such plasticity may underlie complex information processing in eukaryotic cells and suggests a functional advantage in having many sites. Our results follow from the unusual geometry of the steady-state phospho-form concentrations, which we show to constitute a rational algebraic curve, irrespective of n. We thereby reduce the complexity of calculating steady states from simulating 3 x 2(n) differential equations to solving two algebraic equations, while treating parameters symbolically. We anticipate that these methods can be extended to systems with multiple substrates and multiple enzymes catalysing different modifications, as found in posttranslational modification 'codes' such as the histone code. Whereas simulations struggle with exponentially increasing molecular complexity, mathematical methods of the kind developed here can provide a new language in which to articulate the principles of cellular information processing.

Figure 1. General model of multisite phosphorylation with substrate S having n sites, kinase E and phosphatase F

The phospho-forms are denoted S_u, where u is a bit string indicating the absence/presence (0/1) of phosphate. Kinase reactions for n=3 with one phosphorylation per reaction (a, distributivity29) and multiple phosphorylations per reaction (b, processivity29). Phosphatases act similarly in removing phosphates. c, Each enzyme (X=E or F) uses a standard biochemical mechanism but may form multiple products, with associated parameters ($a_u^X,b_u^X,c_{u,v}^X$) for mass-action kinetics. ATP is assumed to be held constant and synthesis and degradation are ignored. d, The 3×2ⁿ differential equations, where $u\stackrel{X}{\to }v$ signifies that X converts S_u to S_v. The same symbol is used for a chemical species and for its concentration. Note that X(S_u) here indicates the product of X and S_u.

Figure 2. Multistability for an n=4 distributive, sequential system, as in Fig. 3a

For parameter values, see Supplementary Table 1. a, Plots of equation (2) for S_tot=10 μM and E_tot=F_tot=2.8 μM. The intersections correspond to the steady states. Filled squares are stable: 1 (red), 2 (black) and 3 (blue); open squares are unstable. Stability was determined by standard methods (Supplementary Information). The inset shows the corresponding phospho-form distributions, following the notation in Fig. 3a. b, Time courses of S₄ reaching its three stable values from initial conditions S₀=αS_tot, S₄=(1 – α)S_tot and X=0 for all other variables, with α chosen randomly from the uniform distribution on [0, 1] (100 samples), determined by simulation (Supplementary Information).

Figure 3. Multistability by kinetic trapping

a, A distributive, sequential system with n=4; E phosphorylates in order, F dephosphorylates in reverse order. The phospho-forms are denoted S_i, i=0,…, 4, where i is the number of phosphates. b, Rate functions for production of S₁ from S₀ by E and of S₀ from S₁ by F are approximated as Michaelis–Menten hyperbolas, with the phosphatase curve to the left and above the kinase curve. If the system is initiated with substrate entirely in S₀ and in excess over both enzymes and saturating them, S₀ will sequester E and produce S₁ at nearly maximal rate (point 1). F, however, will be unoccupied, so that as S₁ increases (grey arrow), phosphorylation and dephosphorylation will balance (point 2). Any leak of S₁ into S₂ can be opposed by F, which is not sequestered. The system will hence reach steady state with substrate remaining predominantly unphosphorylated. If similar conditions are applied to S₃ and S₄, but reversed with respect to E and F, then S₄ can be similarly trapped. The parameter values required for this argument are discussed in the Supplementary Information.

Figure 4. Switching between stable states

a, The system in Fig. 2a with E_tot=F_tot=2.8 μM and S_tot=10 μM is taken in a cycle (grey arrows) using simulation (Supplementary Information). The free kinase is repeatedly changed by a small amount and the system allowed to relax back to a stable state. Starting on the lower branch, corresponding to distribution 1 in Fig. 2a, the system switches abruptly to the higher branch (distribution 3), remains on that branch as E_tot is lowered, and then switches abruptly back down to the lower branch (hysteresis). b, The system with E_tot=F_tot=2.8 μM, S_tot=5 μM has only two stable states (not shown) but occupies three when E_tot is cycled. Three stable states only coexist in a narrow window around the dotted line. The solid lines mark E_tot=2.8 μM.