Dynamic control of catalysis within biological cells

We develop a theory of enzyme catalysis within biological cells where the substrate concentration [S](t) is time dependent, in contrast to the Michaelis-Menten theory that assumes a steady state. We find that the time varying concentration can combine, in a non-linear way, with the ruggedness of the free energy landscape of enzymes (discovered both in single molecule studies and in simulations) to provide a highly efficient switch (or, bifurcation) between two catalytically active states, at a critical substrate concentration. This allows a dynamic control of product synthesis in cell. The concentrations of the constituents (Proteins, ATP, ADP, pH, etc.) in a healthy cell are strictly controlled and this control is dynamic in nature. The survival of the cell is crucially dependent on the several switches which operate with surprising efficiency in turning off and on the synthetic machines in the cell [1]. Synthesis of constituents in a cell is conducted by enzymes, sometime called “molecular machines”. A specific enzyme converts a particular substrate to a particular product by means of a specific chemical transformation [2]. Moreover, several studies have revealed that substrates are produced within a cell in a burst [3]. This may cause a dangerous imbalance in cell as concentration of that particular species can become alarmingly large subsequent to the burst. Enzyme should then start consuming that substrate at a high rate to bring back the concentration close to its physiological range. Once the imbalance is corrected, the machine should consume the substrate at a steady rate to prevent significant lowering of the concentration. A particular enzyme may thus need to act as a substrate concentration dependent switch that operates around a critical concentration of a substrate (physiological concentration). However, how an enzyme performs its amazing task is still shrouded in mystery. A possible scenario is that an enzyme exists in two different conformational states below and above this critical concentration [4]. A biological cell must then have a sensor of substrate concentration that triggers a switch between the two states. Usual description of switch in cells employs multiple enzyme receptor sites and switching between different states occurs by receptor binding [5]. Nonlinear mathematics (such as bifurcation theory) can produce the essential characteristics of such switches [6-8]. For example, in the case of genetic toggle switch in Escherichia coli it is the receptor concentration that acts as the control parameter [5]. We are not aware of any discussion where substrate and/or product concentration act as the control parameters for an enzymatic switch. Our understanding of the substrate concentration dependence of the enzyme kinetics is based on the Michaelis-Menten (MM) theory that assumes a time independent steady state substrate concentration [9]. Several theoretical studies have examined the validity of MM kinetics in the single molecular level [10-13]. The steady state operates under a chemical potential gradient where the in flow of the substrate and the out flow the product are guaranteed. Additionally, the substrate concentration is higher than that of the enzyme. Although the MM kinetics has been amply verified for in vitro experiments both at ensemble and at single molecular level [14], its validity inside the cell remains doubtful because inside the cell substrate concentration is time dependent and also the concentrations of enzyme and substrate are comparable. The relevance of the above issues was made apparent in a recent experiment that revealed that GroEl enzyme that hydrolyzes ATP to ADP works in a different catalytic cycle when the concentration of ATP is high [4]. It was also observed that enzyme did not return to its equilibrium relaxed state after the product (ADP) was released. Instead, it was found to bind to ATP from an intermediate state to continue the reaction cycle with a higher rate. In addition, when ATP concentration became low, then the enzyme relaxed back to the equilibrium R-state where it was found to bind to ATP to continue the reaction cycle with a lower rate [4]. Recent detailed theoretical and very long computational study (~ 1μs) on the catalytic conversion of ATP and AMP to ADP by adenylate kinase (ADK) revealed the existence of a half-open halfclosed (HOHC) intermediate state that can modify the catalytic cycle and accelerate the rate [15]. The intermediate state in an enzyme catalysis bears the similarity with the intermediate state in organic synthesis where the sequence of intermediates dictates the course of the reaction. Existence of such intermediate states in the free energy surface suggests ruggedness of the free energy surface. Some of these ruggedness arises due to water mediated interaction [15]. The intermediate HOHC state was conjectured to facilitate the catalysis by establishing a nonequilibrium steady state [16]. A recent theory of enzyme catalysis envisaged a non-equilibrium conformational cycle in which the active enzyme never needs to return to its native state during the cycle [17-18]. The cycle, which is maintained by steady inflow and outflow of the substrate and product, respectively, is driven of course by the free energy gradient. In this theory, depending on the rate of relaxation of the enzyme conformation after product release, the enzyme can capture a new substrate from an intermediate state and thereby the rate of catalysis can increase. We show below that with suitable generalization, such as the inclusion of the ruggedness of the free energy surface, one can have a molecular level description of a switch where the said mathematical non-linearity arises naturally. We employ a free energy surface for enzyme catalysis as shown in Figure 1. The reaction cycle consists of several sequential steps. (i) The initiation of the reaction cycle which could be activated, (ii) the use of substrate-enzyme interaction to steer both towards reactive geometry, (iii) the catalytic conversion to form the product, (iv) the product release and subsequent enzyme relaxation back towards native state conformation, (v) the substrate capture and continuation of the cycle. Figure 1: The free energy surface of enzyme catalysis. Intermediate state e1 is introduced in the enzyme relaxation surface after the product release. A similar intermediate state a1 is also introduced in the E...S similar to the e1. The usual equilibrium states, e0 and a0, are also present in the present model. Relevant free energy barriers and the reaction steps are shown by arrows. Here ΔG1 determines the residence time of the e1 state and also the substrate concentration dependent switch of the catalysis. The value of ΔG1 also determines the extent of ruggedness in the free energy surface of enzyme fluctuations. We introduce an intermediate state (e1) in the enzyme relaxation surface of the product side (EP surface) in addition to the equilibrium state (e0), and an intermediate state (a1) in addition to the conventional equilibrium state (a0) of the substrate bound enzyme on the substrate site (ES surface). We have introduced the ruggedness in the free energy surface by introducing such intermediates in the spirit of our recent finding of new HOHC intermediate for the adenylate kinase enzyme [15, 16]. Introduction of such an intermediate can modify the catalytic cycle in the following manner: (1) while relaxing back to e0 after the product release (e state), the enzyme now gets trapped in the e1 state. The residence time of the enzyme in that minimum is dependent on the barrier (ΔG1) it experiences in the process of relaxation back to e0. Now, if substrate encounters the e1 state within its residence time, it goes to a1 state directly in the ES surface. From the a1 state it has to surmount a smaller barrier (ΔG3) to reach the downhill induced fit surface which takes the enzyme substrate complex to the reactive geometry. Clearly, this situation can happen at the high substrate concentration limit. The rate of catalysis for such a truncated cycling is high. (2) On the other hand, if substrate does not encounter the e1 state within its residence time, it goes back to e0 state and captures substrate to reach a0 state of the ES surface. To reach the downhill induced fit surface, the a0 state has to surmount two consecutive barriers (ΔG2 and ΔG3 ). Thus, in such a scenario of extended cycling, the reaction rate can become quite slow. Clearly, such a situation appears when the substrate concentration is low. Thus, in our present model we have a crossover between the truncated and extended cycling depending on the substrate concentration and the residence time of the enzyme in the intermediate state e1. According to the scheme shown in Figure 1, we can write the rate equations for different species. Several formalisms have been developed and discussed about motion on a rugged energy landscape [19, 20]. The rate equation for the state e0 can be written as,           0 1 0 1 0 0 0 1 [ ] e e e e e a e dp t k G p t k S p t dt      . However, after the initiation of the reaction, this step can be pre-empted by the establishment of a non-equilibrium steady state that occurs at high substrate concentration, as described above and in the discussion of Figure 1. We show below that such a crossover indeed happens above a certain critical concentration [SC] whose expression is provided later. This feature can be captured by introducing a Heaviside function (H) in the rate equations as,                       0 0 0 1 1