{'fraction_non_alphanumeric': 0.026486302148459703, 'fraction_numerical': 0.005342730476298738, 'mean_word_length': 4.896782841823057, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
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.

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][7][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][11][12][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].  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  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 e 0 can be written as, 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 [S C ] whose expression is provided later. This feature can be captured by introducing a Heaviside function (H) in the rate equations as, Here p i (t) are the time dependent probability of species i and k i→j are the rate of conversion from state i to j. Similar equations are also employed for p a1 (t) and p a0 (t). k φ is given by The mean first passage time (which is the inverse of the rate of the enzyme catalysis (ν)) can be obtained by solving the above equations as, The first part of the left hand side of the above equation provides the rate when substrate concentration is higher and the second part will contribute when the substrate concentration is low. In Eq.3, [S] is an instantaneous substrate concentration. Note that Eq.3 follows essentially from the ruggedness of the free energy surface, shown in Figure 1, and from the kinetic scheme presented above.       We now solve Eq. 3 with time dependent substrate concentration self consistently to get time dependent rate of the catalysis which is shown in Figure 4(a). As the initial substrate concentration is high (subsequent to burst at t =0), the rate readily becomes high (switch at high [S]) and decreases with time slowly. When the substrate concentration reaches a value near the critical value (here, [S C ] ≈ 3.7 in reduced unit for the parameters used), the rate suddenly decreases. This gives rise to a switch in the time dependence of the rate of catalysis. The above calculation also provides the time dependent substrate concentration which is shown in Figure   4(b).
[S](t) decreases rapidly (after the burst) till it reach its critical value and then it starts to decrease slowly. These results indicate the following: (1) when the substrate concentration becomes alarmingly high, enzyme starts to work at a high rate to consume the substrate faster [switch at high [S]). (2) When the substrate concentration reaches its physiological limit (critical concentration), enzyme starts to work at a lower rate (switch at low [S]) and the substrate concentration does not further change rapidly. Thus, such an enzymatic switch is a necessary mechanism for the cell to maintain the physiological concentration of its constituents.
While we are not aware of any previous theoretical study that discusses the possibility of a switch driven by time dependent substrate concentration, the results derived here from the theoretical model of dynamic control (and switch) are in good agreement with experiment models and simulation studies. Most importantly, our model offers perhaps the simplest example of the switch and is based on time varying substrate concentration: the switch is turned on when the concentration of the substrate is large. A similar switch can be built that is based on product concentration. When the product concentration becomes large, the product release step can become slow, thus again switching off the molecular machine. Thus, the mechanism proposed and developed here can be incorporated within a network within the cell where different molecular machines microscopically "talk" with each other through time varying substrate and product concentrations. In such a network, product from one machine is the substrate or fuel for a second machine, and so on.
The merit of the present model is that it reveals that fact how enzyme uses conformational landscape of its own to maintain a critical concentration of important substrates inside the cell.
Although, the present model is also simple, it essentially describes the substrate concentration dependent switching of the enzymatic activity in a novel way.