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
(5)
where Vk(ADP) and Vp(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:
(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:
(7)
This is a requirement that cannot generally be met, because the input signal ADP changes Vk and Vp in opposite directions (except for a single value of ADP, Vk and Vp 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]
(8)
Where Ki,ppi = 160 μM is the inhibition constant [17] and Vp0(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 (Ki,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
(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,
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:
(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
(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 Vk < Vp0, 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.
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