Robust Control of PEP Formation Rate in the Carbon Fixation Pathway of C₄ Plants by a Bi-functional Enzyme

A mechanism based on bifunctional enzyme avidity and product inhibition suggests robustness of PEP formation rate

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.

The avidity effect in RP action

We denote the non-phosphorylated form of PPDK by PPDK₀, the phosphorylated form at the His residue by PPDK1 and the doubly phosphorylated form at the His and Thr residues by PPDK₂. Only PPDK₁ is active and catalyzes the production of PEP.

The bifunctional enzyme RP has two domains, one for kinase and the other for phosphatase activity [20, 21]. This two-domain structure, together with the tetrameric form of its substrate PPDK, provides for a cooperative binding effect known as the avidity effect. Avidity results when one domain of RP, the kinase domain, binds a PPDK₁ subunit and the other domain binds a PPDK₂ subunit on the same tetramer.

We name the situation where RP simultaneously binds two PPDK subunits as the ternary complex, [PPDK₁ RP PPDK₂]. The situation where RP binds only one domain is termed a binary complex. The binary complexes are [RP PPDK₁] and [RP PPDK₂]. Thus, at steady-state, the total RP kinase activity equals the RP phosphatase activity, and includes the contribution of both binary and ternary complexes

$\begin{array}{c}{\mathsf{\text{V}}}_{\mathsf{\text{k}}}\left(\mathsf{\text{ADP}}\right)\left(\left[\mathsf{\text{RP PPD}}{\mathsf{\text{K}}}_{\mathsf{\text{1}}}\left]+\right[\mathsf{\text{PPD}}{\mathsf{\text{K}}}_{\mathsf{\text{1}}}\mathsf{\text{RP}}\mathsf{\text{PPD}}{\mathsf{\text{K}}}_{\mathsf{\text{2}}}\right]\right)=\\ ={\mathsf{\text{V}}}_{\mathsf{\text{p}}}\mathsf{\text{(ADP)}}\left(\left[\mathsf{\text{RP}}\mathsf{\text{PPD}}{\mathsf{\text{K}}}_{\mathsf{\text{2}}}\left]+\right[\mathsf{\text{PPD}}{\mathsf{\text{K}}}_{\mathsf{\text{1}}}\mathsf{\text{RP}}\mathsf{\text{PPD}}{\mathsf{\text{K}}}_{\mathsf{\text{2}}}\right]\right)\end{array}$

(5)

where V_k(ADP) and V_p(ADP) are the specific catalytic activities of the two domains of RP. These rates depend on the input ADP [17].

Due to the avidity effect, however, the ternary complex is highly favored relative to binary complexes. Once RP binds one subunit of PPDK, for example PPDK₁, the effective local concentration of a neighboring subunit (PPDK₂) is increased. As a result, the on-rate for the second binding is very high (typical avidity effects show an on rate that is 100 times or more larger than the first binding rate [4, 22]). Unbinding is rare, because both subunits need to unbind at the same time for RP to leave the tetramer.

Avidity therefore ensures that, as long as both PPDK₁ and PPDK₂ forms are present on the same tetramer, the ternary complex is the prevalent complex in the system (see Figure 2). As a result, both phosphorylation and de-phosphorylation catalyzed by RP occur mainly in the ternary complex. This applies also in a more detailed model, presented in the last section of the results, that takes into account the spatial organization of the three possible states of PPDK subunits along the tetramer. Thus, we assume that, as a first approximation, we can neglect the binary complexes in Eq.(5), to find that the condition for steady-state is equality between the rates catalyzed by ternary complexes:

${\mathsf{\text{V}}}_{\mathsf{\text{k}}}\left(\mathsf{\text{ADP}}\right)\left[\mathsf{\text{PPD}}{\mathsf{\text{K}}}_{\mathsf{\text{1}}}\mathsf{\text{RP}}\mathsf{\text{PPD}}{\mathsf{\text{K}}}_{\mathsf{\text{2}}}\right]={\mathsf{\text{V}}}_{\mathsf{\text{p}}}\left(\mathsf{\text{ADP}}\right)\left[\mathsf{\text{PPD}}{\mathsf{\text{K}}}_{\mathsf{\text{1}}}\mathsf{\text{RP}}\mathsf{\text{PPD}}{\mathsf{\text{K}}}_{\mathsf{\text{2}}}\right]$

(6)

When the ternary complex level is non-zero, one can cancel it out from both sides of the equation. This means that steady-state requires equal specific kinase and phosphatase rates for the bifunctional enzyme RP:

${\mathsf{\text{V}}}_{\mathsf{\text{k}}}\left(\mathsf{\text{ADP}}\right)={\mathsf{\text{V}}}_{\mathsf{\text{p}}}\left(\mathsf{\text{ADP}}\right)$

(7)

This is a requirement that cannot generally be met, because the input signal ADP changes V_k and V_p in opposite directions (except for a single value of ADP, V_k and V_p are generally unequal). Thus, steady state requires an additional layer of regulation. We next describe an effect due to product inhibition, which can satisfy the steady-state condition, and turns out to provide robustness.

Tuning of RP velocities can be achieved through a product-inhibition feedback loop

Note that the products of the auto-kinase reaction of PPDK, AMP and PPi, are also the products of the RP reactions: AMP is the product of the kinase reaction, and PPi the product of the phosphatase reaction. In the present view, these features can help to form a robust mechanism, because they provide a feedback loop between PPDK and RP activities. This feedback is due to the phenomenon of product inhibition [19] of RP. The phosphatase activity of RP has been found to be inhibited by its product PPi, following a Michaelis-Menten like inhibition curve [17]

${\mathsf{\text{V}}}_{\mathsf{\text{p}}}\left(\mathsf{\text{ADP}},\mathsf{\text{PPi}}\right)={\mathsf{\text{V}}}_{\mathsf{\text{p}}0}\left(\mathsf{\text{ADP}}\right)∕\left(\mathsf{\text{1}}+\left[\mathsf{\text{PPi}}\right]∕{\text{K}}_{\text{i},\text{PPi}}\right)$

(8)

Where K_i,_ppi = 160 μM is the inhibition constant [17] and V_p0(ADP) is the maximal phosphatase velocity. Thus, the more PPi in the cell, the lower is the phosphatase activity of RP. Experiments suggest that the kinase reaction of RP is not measurably inhibited by the second product AMP (K_i,_AMP > 2 mM, [17]).

Since PPi is produced by PPDK, and inhibits RP, it can link these two enzyme activities. For this to happen, however, the concentration of PPi in these cells must be determined mainly by PPDK, and not by the hundred or so other reactions that produce PPi [23]. The situation in these plant cells might be special, however, because of the huge amount of PPDK enzyme (7-10% of total protein). We therefore assume that the main production source of PPi is the PPDK auto-kinase activity, and neglect to a first approximation all other PPi sources (see also additional file 2). The concentration of PPi in such a case is given by the balance of its production rate by the PPDK₀ auto-kinase reaction, F₁(ATP,Pi,PPi), and its degradation at rate α

$\mathsf{\text{d}}\left[\mathsf{\text{PPi}}\right]∕\mathsf{\text{dt}}={\mathsf{\text{F}}}_{\mathsf{\text{1}}}-\alpha \left[\mathsf{\text{PPi}}\right]$

(9)

Solving this results in a steady-state concentration of PPi that is proportional to the production rate from the auto-kinase reaction (F₁), [PPi] = F₁/α. This is important because at steady-state each auto-kinase reaction corresponds to one PEP formation reaction: the phosphate is transferred from PPDK onto pyruvate to produce PEP. Because of this stoichiometric relationship, the system output, PEP formation rate F₂, is equal to the production rate of PPi from the auto-kinase: F₂= F₁. These considerations link the PEP formation rate, F₂, to the PPi concentration,

${\mathsf{\text{F}}}_{\mathsf{\text{2}}}=\alpha \left[\mathsf{\text{PPi}}\right]$

(10)

Using this relation in Eq.(8), we see that product inhibition of RP by [PPi] leads to the following connection between the systems output F₂ and the RP phosphate rate:

${\mathsf{\text{V}}}_{\mathsf{\text{p}}}\left(\mathsf{\text{ADP}}\right)={\mathsf{\text{V}}}_{\mathsf{\text{p}}0}\left(\mathsf{\text{ADP}}\right)∕\left(\mathsf{\text{1}}+{\mathsf{\text{F}}}_{\mathsf{\text{2}}}∕{\mathsf{\text{F}}}_0\right)$

(11)

Where F₀ = α Ki,PPi. This closes a negative feedback loop: the higher the PEP formation rate F₂, the lower the phosphatase activity of RP, and thus the more PPDK in its inactive form PPDK₂, leading to lower PEP formation rate (see Figure 3). This loop leads the PEP formation rate to a point at which the RP kinase and phosphatase activities are equal (Eq.(7)). Using Eq.(11), we find that this steady state PEP formation rate is

${\mathsf{\text{F}}}_{\mathsf{\text{2}},\mathsf{\text{st}}.\mathsf{\text{st}}}={\mathsf{\text{F}}}^{*}=\alpha {\mathsf{\text{K}}}_{\mathsf{\text{i}},\mathsf{\text{PPi}}}\left({\mathsf{\text{V}}}_{\mathsf{\text{p}}0}\left(\mathsf{\text{ADP}}\right)∕{\mathsf{\text{V}}}_{\mathsf{\text{k}}}\left(\mathsf{\text{ADP}}\right)-\mathsf{\text{1}}\right)$

(12)

This is the main result of the present analysis. The output formation rate F* does not depend on the concentrations of the proteins in the system, RP and PPDK. It also does not depend on any of the substrate metabolites, ATP, pyruvate, PEP and AMP. The formation rate is thus robust to these potentially fluctuating concentrations as been also suggested by studies in leaves and isolated chloroplasts showing no clear relation between PPDK activity and changes in ATP, AMP, pyruvate and PEP levels (reviewed in [10] and references therein, see also additional file 2). Despite this robustness, the output rate is controlled by the input signal ADP, which corresponds to light levels.

The magnitude of the output (PEP formation rate) in this mechanism is given by the product of the PPi product-inhibition constant and the PPi degradation rate, F₀ = α Ki,PPi. We note that pyrophosphatases are abundant in the chloroplast [24], providing a fast hydrolysis specific activity of 40 μmol/mg chl/min [25], yielding α ≈100 [1/sec]. Since Ki,PPi = 160 μM [17] one finds a rate of about F₀ = 10⁸ reactions/second per chloroplast (for chloroplast of size 20 μm³ [26]). This rate magnitude makes sense: the C₄ cycle in these plant cells assimilates about 10⁷-10⁸ carbon atoms in the form of CO₂ per second per chloroplast at daylight [27, 28] (see additional file 2 for more details).

Limits of robustness

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).

A model for the spatial arrangement of PPDK subunits based on avidity predicts a bimodal distribution of phosphorylated and unphosphorylated tetramers

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.

Mathematical model of the ternary complex avidity

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.

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].

Solution of the product inhibition equation

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

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].

Spatial model of PPDK configurations

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 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.

Analytical results of the major configurations' occupation at steady-state

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.

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.

Ternary to Binary reactions ratio is inversely dependent on the number of protein's subunits

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.

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.