We present a mechanism for robustness in the system based on its known biochemical features. The outline is as follows: we first note that the tetramer structure of PPDK makes possible an avidity effect, in which RP primarily acts when it is bound at the same time to two different monomers on the same tetramer. We then show that this avidity effect allows the system to reach steady-state only if the specific rates of the kinase and phosphatase reactions of RP are exactly equal. Finally, we note that such tuning of specific rates is made possible by a feedback loop, in which the rate of PEP formation affects RP rates by product-inhibition (through the shared metabolite pyrophosphate). The upshot is that the PEP formation rate (the output of the system) depends only on the input signal (ADP, which corresponds to light level), and not on any of the protein levels (PPDK, RP), levels of metabolite substrates (pyruvate, ATP) or on PPDK catalytic rate. The full set of equations of the mechanism is shown in additional file 1. The following description aims to allow an intuitive understanding of the mechanism.

We also studied the conditions in which robustness might break down. The model suggests three cases: The first potential condition for loss of robustness is when there is not enough total PPDK enzyme or substrates to provide the robust rate F* of Eq.(12). The second includes conditions of very low or very high input signal, in which the binary complexes in Eq.(5) cannot be neglected, and avidity is no longer a dominant effect. The third condition for loss of robustness occurs when total PPDK levels are extremely high such that its activity cannot be regulated due to shortage in the phosphorylation substrate (ADP levels). We now briefly analyze these conditions.

The first type of conditions in which robustness does not occur is when there is not enough total PPDK enzyme or substrates (ATP, pyruvate) to provide the robust PEP formation rate F* given by Eq.(12). For example, if substrate or PPDK levels are zero, one must have F₂ = 0. Solution of the model shows that when one of these factors (total PPDK, pyruvate or ATP levels) goes below a threshold concentration (equal to its minimal concentration needed to reach F*), all of PPDK becomes active (PPDK₂ = 0). The formation rate F₂ is then linear in PPDK1, F₂ = V1(pyr) PPDK1. In this state, the rate depends on protein and metabolite levels and robustness is lost. As soon as PPDK and/or substrate levels become high enough to reach F*, robustness is restored (see Figure 4).

The second case for loss of robustness is extreme input levels in which the binary complexes are not negligible compared to ternary complexes. Avidity requires that PPDK exist on the same tetramer in both PPDK1 and PPDK₂ forms. However, in extreme high or low signal (ADP) levels, this does not apply. In these conditions, one can no longer neglect the effects of binary complexes (see Methods and additional file 2). At very low ADP levels (very high light), most PPDK is active and PPDK₂ monomers are rare. Ternary complexes are scarce because they require PPDK₂.

We estimate that robustness begins to erode at light levels below 50 μE m^-2 s^-1 or above 800 μE m^-2 s^-1, which is also the mean photosynthetic photon flux at daylight 28 29 . Thus robustness is found between an upper and lower bounds on the light input (and its corresponding ADP encoding), as illustrated in Figures 2 and 4.

Robustness also breaks down at an extreme case when total PPDK levels exceed ADP concentration (PPDK _T >> ADP), a condition that physiologically cannot be met due to the very high levels of this protein. In this case, cellular ADP levels are too low to allow further phosphorylation of the excess PPDK₁. Consequently, the rate of PEP formation will be linearly dependent on PPDK total amounts (see Figure 1b, high end of the x axis and additional file 2).

We also note that to be feasible, the robust mechanism must admit a positive and stable solution. Exact solution of the model shows that this corresponds to the condition V_k < V_p0, namely that the RP kinase rate is smaller than the phosphatase maximal rate (the rate in the absence of inhibition).

Finally we analyze the detailed configurations of PPDK states within PPDK tetramers, when the robust mechanism is active. The robust mechanism involves the RP cycle catalyzed primarily by RP bound to two adjacent subunits of PPDK, one in PPDK₂ form and the other in PPDK₂ form. The abundance of this ternary complex relative to binary complexes is due to the avidity effect.

When RP carries out a reaction, it changes the state of one of the two subunits that it binds: changing PPDK₁ to PPDK₂ or vice verse. It thus converts adjacent PPDK₁- PPDK₂ subunits either to two adjacent PPDK₁ subunits, or two adjacent PPDK₂ subunits.

The action of RP therefore tends to convert neighboring subunits that have different forms to the same form. Analyzing this in a detailed model that tracks the different configurations of tetramers (see Methods), we find that the dynamics reaches a steady-state in which the configuration distribution resembles a bimodal distribution. In this distribution, tetramers tend to be made of all PPDK₁ or all PPDK₂ subunits (Figure 5). These forms are slowly converted to other forms by RP binding to a single monomer (binary complex). The rarest forms are those with adjacent PPDK₁- PPDK₂ states, arranged in a "checkerboard" pattern. A quantitative analysis of the configuration probability distribution and its effect on the ratio of ternary to binary reactions is presented at the Methods section.

We also studied the effect of a three-state model on the different configurations, with 3 possible states for PPDK subunits, namely PPDK₀, PPDK₁ and PPDK₂. We find that for the system to attain robustness it is beneficial that the two steps of the phospho-transfer have different rates. Only if the auto-phosphorylation of PPDK is faster than the phospho-transfer to pyruvate, the majority of the PPDK pool will transition between the PPDK₁ and PPDK₂ states and the ternary complex will dominate the modification reactions. Otherwise, the majority of the configurations will be in the PPDK₀ state which hampers the probability for a ternary complex to exist. In-vitro measurements suggest that the auto-phosphorylation reaction is 1.5 faster than the phospho-transfer reaction 30 . We find that this is sufficient for the avidity reactions to dominate the process, and for robustness to result.

We used mass-action kinetics to describe the model illustrated in Figure 2:

d [ PPD K 1 RP ] ∕ dt = kon1 PPD K 1 RP - koff1 + V k + kon2 p1 [ PPD K 1 RP ] + koff2 PPD K 1 RP PPD K 2 = 0

d PPD K 2 RP ∕ dt = kon4 PPD K 2 RP - ( koff4 + V p + kon3 p2 )   PPD K 2 RP + koff3 [ PPD K 1 RP PPD K 2 ] = 0

d [ PPD K 1 RP PPD K 2 ∕ dt = kon2 p1 PPD K 1 RP + kon3 p2 PPD K 2 RP ] - ( koff2 + koff3 + V k + V p ) PPD K 1 RP PPD K 2 = 0

Here p1 denotes the probability for a PPDK₂ subunit near a bound PPDK₁ subunit, and p2 is the same for a PPDK₁ subunit near a bound PPDK₂ subunit. The simplest model assumes that p1 and p2 are the fractions of the corresponding monomers

p1 = PPD K 2 ∕ PPD K T

p2 = PPD K 1 ∕ PPD K T

A more detailed model that takes into account the spatial configurations of PPDK₀, PPDK₁ and PPDK₂ in the tetramer is provided in the next section. The detailed model shows similar results for robustness, and makes further predictions on the correlations of the states of adjacent monomers.

These equations were solved analytically using the fact that PPDK concentration is much higher than RP levels (more than a 100-fold higher 13 33 ). The avidity effect allows us to assume that the on-rate for RP bound to one monomer to bind an adjacent monomer on the same tetramer is very large, due to the increased local concentration (kon2, kon3 >> kon1 PPDKT, kon4 PPDKT). Also, off-rates are assumed to be much faster than enzymatic reactions rates as is the case for most enzymes (in-vivo experiments suggest that full activation/de-activation occur on the scale of 10-60 minutes, therefore phosphorylation/de-phosphorylation rates are in the range of 1-10 [1/sec], compared to koff rates on the scale of 1 msec 13 20 11 thus V_p, V_k << koff1, koff4). Analytical solution of the model was obtained using Mathematica 7.0.

A lower bound for the ratio of ternary to binary complexes is obtained by solving Eq.(15) at steady-state. We denote B1 as the binary complex [PPDK₁ RP], B2 as the binary complex [PPDK₂ RP] and T as the ternary complex [PPDK₁ RP PPDK₂]. Also, for simplicity we assume symmetrical rates on both branches (kon2 = kon3, koff2 = koff3)

( 2 koff2 + V k + V p ) T = kon2 (p1 B1 + p2 B2)

Therefore, the fraction of ternary to binary complexes is bounded from below by

T / ( B1 + B2 ) ≥ kon2 / ( 2 koff2 + V k + V p } )  min(p1 , p2)

Since kon2 is very large compared to koff2 due to the avidity effect 22 , kon2/koff2 = A >> 1, and one can neglect V_k,V_p compared to koff2 13 11

T ∕ ( B1 + B2 ) ≈ A min(p1 , p2) > > 1

This prevalence of the ternary complex breaks down only when the probabilities p1 or p2 become small, on the order of 1/A where A is estimated to be on the order of 100 11 22 34 .

To describe the output formation rate F as a function of the input (ADP levels) we used Eq. (5):

V k ( ADP ) ( B1 + T ) - V p ( ADP ) ( B2 + T ) = 0

where V_k and V_p are effective phosphorylation and de-phosphorylation rates, B1 = [ PPDK₁ RP] and B2 = [ PPDK₂ RP] are the concentrations of the binary complexes and T = [ PPDK₁ RP PPDK₂] is the ternary complex concentration. Since V_k is activated by ADP and V_p is inhibited by ADP in a Michaelis-Menten fashion 17 35 , we take the dependence on ADP levels to be

V k ∕ V p = V k 0 ∕ V p 0 ADP

where V_k0 and V_p0 are effective rate constants dependent on enzyme catalytic rate and on/off rates.

Product-inhibition of PPi is formulated most generally as 19 :

Vp = ( V p 0 - V ′ p PPi ∕ K i2 , PPi ) ∕ ( 1 + PPi ∕ K i , PPi )

where the catalytic rate and the binding are both lowered due to the presence of the reaction product. For clarity, we assume that the product inhibition is of a competitive type 17 , where the product occupies the catalytic site thus preventing catalysis. The equations for the rates then simplify to

V p = V p 0 ∕ ( 1 + PPi ∕ K i , PPi )

For Figure 4, these equations were solved numerically for different concentrations of PPDK total amount using Mathematica 7.0.

We finally note that perfect robustness (complete insensitivity) to all protein and metabolites is an idealized feature. One may ask whether it persists if one adds additional reactions which have been neglected due to their small relative rates. We find (see additional file 2), that adding such reactions (e.g. the contribution of the binary complexes, the contribution of other reactions that make AMP and PPi) preserves approximate robustness: if the rates of these reactions are on order of a small number ε relative to the corresponding reactions above, sensitivity to proteins and metabolites is no longer strictly zero but is small, on the order of ε. We also note that the dependence of V_p0 on Pi levels is neglected in this discussion due to the high and buffered Pi levels in the cell, making this metabolite unlikely to fluctuate as much as other metabolites 36 37 .

We developed a model in order to study the binding of RP to the four subunits of PPDK. For a ternary complex to form, RP must bind a PPDK₁ and a PPDK₂ that are neighboring subunits. It then can catalyze either phosphorylation or de-phosphorylation. Each PPDK subunit has three conformations possible: PPDK₀, PPDK₁ and PPDK₂. Taking into account the symmetries of the tetramer there are 21 possible configurations.

The relative occupancy of the different configurations was calculated using a Master equation model, with transitions between configurations carried out by the binding of the enzyme RP, to a single domain or two adjacent domains. The single-domain binding events are essential to prevent the system from becoming stuck in an all PPDK₁ or all PPDK₂ state. The probability of state i is P(i), and the transition rate to state j is wij, and

d P ( i ) d t = ∑ j ≠ i w j i P ( j ) - ∑ j ≠ i w i j P ( i )

The transition probabilities were calculated based on the number of adjacent PPDK₀, PPDK₁, PPDK₂ and PPDK₁ PPDK₂ subunit pairs in the configuration. Each configuration has a probability μ to phosphorylate or de-phosphorylate any one of its subunits by single-subunit binding of RP, or RP can bind adjacent PPDK₁ and PPDK₂ subunits (if they exist in that configuration) and to either phosphorylate the PPDK₁ subunit with a probability η1 or de-phosphorylate PPDK₂ with a probability η2. Since the ternary complex is long-lived, the reaction almost always occurs before unbinding, and thus η1 and η2 approach unity. Also, in the binary complex, the probability for a reaction to occur before unbinding is the ratio of RP catalytic rate to the unbinding rate, so that:

μ / η 1 , 2 = 1 / 2(V k / ( koff1 + V k ) + V p /(koff4 + V p ) )

The probability for an auto-phosphorylation reaction of a PPDK₀ subunit is denoted by δ1 and the phospho-transfer reaction by δ2. Therefore, in order to have accumulation of PPDK₁ and PPDK₂ subunits, δ1 should be greater than δ2.

Solving the Master equation yields the fraction of each configuration at steady-state. We find that the avidity mechanism favors clustering of the PPDK₁ and PPDK₂ subunits. Thus, PPDK₁ and PPDK₂ subunits tend to be maximally spatially separated.

The following simplified model allows for an analytical estimate of the configurations distribution at steady-state for arbitrary protein's size. As shown from the solution of the Master equation (see previous section), the dominant configurations are ones where the modified and unmodified subunits cluster together and are phase separated. Therefore we consider only transitions between these states. The probability for a certain configuration state is denoted by its amount of modified subunits, namely, N(0) is the probability for the configuration where all subunits are unmodified, N(1) is the probability for a configuration with one modified subunit and so on. We further assume that for all states which are not fully modified or unmodified, reactions from the ternary complex are dominant and thus neglect binary reactions from these configurations. The modification reaction rate from a ternary complex is denoted by η1 and de-modification reaction rate from a ternary complex is denoted by η2. Similarly, modification and de-modification reaction rates from a binary complex are denoted by ε1 and ε2 respectively.

The model yields the following set of equations:

d N(0) ∕ dt = - n ε 1 N(0) + 2 η 2 N(1) = 0

d N(1) ∕ dt = n ε 1 N(0) + 2 η 2 N(2) - 2( η 1 + η 2) N(1) = 0

d N(2) ∕ dt = 2 η 2 N(3) + 2 η 1 N(1) - 2( η 1 + η 2) N(2) = 0 …

dN(n) ∕ dt = - n ε 2 N(n) + 2 η 1 N(n - 1) = 0

The general solution is easily admitted. When one assumes that ε1 = ε2 = ε and that η1 = η2 = η, the solution reduces to:

N(0) = N(n) = η ∕ ( 2(n - 1) ε + 2 η )

N(i) = ε ∕ ( ( n - 1 ) ε + η ) , i = 1 , 2 … n - 1

It is thus evident that the ratio between the boundary states to the 'bulk' (i.e. all configurations with partial number of modified subunits) is of order ε/η. This ratio can be viewed as the energetic cost of shifting the boundary between the two domains of modified and unmodified subunits. Also, it suggests (as the Master equation solution indeed indicates) that the neglected states with two or more boundaries between domains are of order (ε/η)^2 and higher, depending on the number of domains.

The avidity effect stems from the ability of the bi-functional enzyme to bind neighboring modified and unmodified subunits. The modification or de-modification reactions are thus limited to proteins that have mixed pairs of modified and unmodified subunits. Here we solve a toy model of the avidity process to assess the dependence of the ratio of ternary to binary reactions on the number of protein subunits.

Following the analysis of configuration states (see above section), we assume that most proteins are phase separated (where a sequence of modified subunits is followed by a sequence of unmodified subunits). Hence, the number of modified subunits is characteristic of the protein state. We also assume that transitions between states with mixed subunits (modified and unmodified) are committed from a ternary complex due to the enhanced local concentration caused by the avidity effect. Therefore, modification and de-modification reactions occur only from the "edge" states when the protein's configuration is either fully modified or unmodified.

We define the mean number of steps to reach one of the boundaries (all modified/all unmodified) as M(i), meaning M(1) is the mean number of steps to reach the boundary from the state of one modified subunit, M(2) is the mean number of steps to reach the boundary from a two modified subunits state and so on up to M(N-1). Each state has a probability p to commit modification and (1-p) to commit de-modification.

Without loss of generality, we can assume that a binary reaction occurred from the 'all unmodified' configuration. Then, M(1) will reflect the ratio between ternary to binary reactions, assuming that ternary reactions have a much more faster time scale than binary reactions (due to avidity, the unbinding of the ternary complex is orders of magnitude slower). To calculate M(1) we solve a regression model, where each state is derived from the state next to it. For example, when the current state is one modified subunit, the mean number of reactions to reach one of the boundaries, M(1), is given by a probability (1-p) to reach the boundary by a de-modification (yielding the 'all unmodified' state) while with probability p the mean number of steps will be one plus the mean number of reactions from the state with 2 modified subunits. The equations to solve are thus:

M(1) = ( 1 - p ) + p(1 + M ( 2 ) )

M(2) = ( 1 - p ) ( 1 + M(1) ) + p(1 + M(3)) …

M(N - 1) = ( 1 - p ) ( 1 + M(N - 2) ) + p

This model can be solved analytically. For concreteness, we take p = 1/2 (a state where modification and de-modification are equally probable, i.e. near the system's steady-state). Then, M(1) simplifies to

M(1) = N - 1

Therefore, near the steady-state the ratio between ternary to binary reactions is proportional to the protein's number of subunits. To confirm this result, we run Monte-Carlo simulations where the number of proteins is two orders of magnitude larger than the number of bi-functional enzymes. We further assumed that there is an equal probability for modification and de-modification and that only states of fully modified and fully unmodified can react from a binary complex. The numerical results indeed show that the ratio between ternary and binary reactions is proportional to the number of protein's subunits and goes as (N-1) where N is the protein's subunits number.