A probabilistic model to describe the dual phenomena of biochemical pathway damage and biochemical pathway repair
aa r X i v : . [ q - b i o . O T ] A p r A probabilistic model to describe the dualphenomena of biochemical pathway damage andbiochemical pathway repair.
Anirban Banerji
Bioinformatics Centre, University of Pune, Pune-411007,Maharashtra, India.
E-mail address : [email protected]
Abstract
Biochemical pathways emerge from a series of Brownian collisions be-tween various types of biological macromolecules within separate cellularcompartments and in highly viscous cytosol. Functioning of biochemicalnetworks suggests that such serendipitous collisions, as a whole, resultinto a perfect synchronous order. Nonetheless, owing to the very natureof Brownian collisions, a small yet non-trivial probability can always beassociated with the events when such synchronizations fail to emerge con-sistently; which account for a damage of a biochemical pathway. Therepair mechanism of the system then attempts to minimize the damage,in the pursuit to bring restore the appropriate level of synchronization be-tween reactant concentrations. Present work presents a predictive prob-abilistic model that describes the various facets of this complicated andcoupled process(damaging and repairing). By describing the cytosolicreality of Brownian collisions with Chapman-Kolmogorov equations, themodel presents analytical answers to the questions, with what probabilitya fragment of any pathway may suffer damage within an arbitrary inter-val of time? and with what probability the damage to a pathway can berepaired within any arbitrary interval of time?
Biochemical pathways come to existence due to series of concentration-dependentcollisions among various species of biological macromolecules that constitute apathway. Traditionally, the time evolution of biochemical pathways is desrcibedby a set of coupled (first order) differential equations that stem from law ofmass action and the information regarding concentrations of each species. Lawof mass action is an empirical law that connects reaction rates with molecu-lar component concentrations through a simple equation. Once provided with1he information of initial molecular concentrations, law of mass action presentsa complete picture of the component concentrations at all future time points(Espenson, 1995). Although popular, this approach (based on law of mass ac-tion) assumes that process of initiating and sustaining the chemical reactions iscontinuous and deterministic (Cox, 1994). As one studies smaller and smallersystems, the validity of a continuous approach becomes ever more tenuous and itbecomes clear that in reality, chemical reactions are innately stochastic (and notcontinuous) in nature. One alsorealizes that reactions occur as discrete eventsresulting from random (and not deterministic) molecular collisions (Gillespie,1977; McAdams and Arkin, 1999; Gibson, 2000; Golightly and Wilkinson, 2006).The stochastic approach attempts to describe this inherent random nature ofmicroscopic molecular collisions to construct a probabilistic model of the reac-tion kinetics (Resat et al., 2001; Qian and Elson, 2002). This approach is thussuited to the modeling of small, heterogenous environments typical of in-vivoconditions (Kuthan, 2001). Such intrinsically stochastic nature of biochemicalreactions have profound implications in many spheres of biology. For example,in the paradigm of molecular binding and chemical modifications, stochastic col-lisions give rise to temporal fluctuations and cell-to-cell variations in the numberof molecules of any given type; they mask genuine signals and responses andfurthermore, contribute critically to the phenotypic diversity in a population ofgenetically identical individuals (Raser and O’Shea, 2004, Maamar et al. 2007).Biochemical pathways are structures that depend crucially upon accurate syn-chronizations between concentration profiles. But stochastic collisions, by theirvery nature, are probabilsitic (Calef and Deutch, 1983). Furthermore, at themolecular level, random fluctuations are inevitable; with their effect being mostsignificant when molecules are at low numbers in the biochemical system (Turneret al., 2004). Therefore, the event of a biochemical pathway malfunctioningcan well be attributed to a failure to ensure synchronization between variousmacromolecular concentrations (Magarkar et al., 2011); which in turn, can beattributed to the inherently probabilsitic nature of the stochastic collisions. Al-though these (serendipitous) Brownian collisions account for the emergence ofintricate and exquisite order in biochemical reactions in most of the cases, theprobability of their failing to achieve the same is cannot merely be trivial. Piv-otally important to stochastic modelling is the realization that molecular reac-tions are essentially random processes and therefore, it is impossible to predictwith complete certainty the time instance at which the next reaction withinany volume may take place (Turner et al., 2004). In macroscopic systems,in the presence of a large number of interacting molecules within the confine-ment of a cellular compartment, the randomness of this behaviour averages out;hence the gross macroscopic state of the system appears to be deterministicand predictable (Minton, 1993; Ahn et al., 1999; Ellis, 2001). However, whilestudying the same from bottom-up approach, one cannot resort to macroscopicdeterminism observed at the limiting case (high concentration of the interactingmacromolecules, highly viscous cytoplasmic fluid, etc.)(Gillespie, 1977; Rao andArkin, 2003). Thus, modeling biochemical reactions from bottom-up perspec-2ive needs to take into account the probability-driven nature of macromolecularinteractions.The present work assumes that owing to inherently probabilistic nature ofthe collisions amongst macromolecules, the adequate level of synchronizationamongst interacting species (that is required to ensure the emergence of macro-scopic deterministic profile) - will fail at times. We hypothesize that it is due tothe failure to ensure the appropriate extent of synchronization that a (fragmentor the entire) biochemical pathway will fail to function. A (fragment or theentire) pathway with such incorrect synchronization is referred to as ’damaged’pathway, in the present work. Though evolution has given rise to robustness ofbiochemical pathways, it is difficult to assume that any arbitrarily chosen path-way will always be functioning with exactly the expected level of optimality. -Though this seems intuitive to appreciate, one will fail to find either a theoreticalor an empirical answer to any of the two questions; one, how many times, withina given interval of time, a pathway (or any section of it) suffers from damage?Two, how soon will the damaged section of the damaged pathway be repaired?etc.. An easy approach to these simple questions may suggest that the answer tothe aforementioned question will be, first: pathway-specific, two: time-intervalspecific, three: organism-specific. However, since evolution tends to reuse thetried-and-tested mechanisms, there is reason to expect that the answers to theaforementioned questions may not be case-specific but general. Therefore, froma ageneral perspective, the present model attempts to quantify one: the prob-ability with which a pathway (or a fragment of it) will malfunction within anyarbitrarily chosen time interval, suitable to observe such event; and two: theprobability with which the damaged pathway (or the damaged fragment of it)will be repaired within any arbitrarily chosen time interval, suitable to observesuch event. Though attempts of probabilistic modeling of biochemical systemsare not entirely commonplace, some previous attempts in the similar lines canbe found in (Hume, 2000; Elowitz et al., 2002; Golightly and Wilkinson, 2006).The present work studies two cases; first-case, when the damaged fragmentof P is detected immediately and repairing of this fragment of P starts withoutany delay, second-case, when the damaged fragment of P is detected after acertain time lag and repairing of this fragment of P , accordingly, starts with adelay. Though no concrete piece of data either supports or contradicts the firstcase; the facts that, one: underlying mechanism of the pathway functioning isrooted in Brownian collisions and therefore is often unreliable, and two: eventhough a pathway functions due to series of favorable but essentially serendip-itous collisions, we do not suffer from too many instances of pathway damage- suggests that probably, after the damage of a fragment of a pathway (whenconcentrations of consecutive species of macromolecules (that constitute thisfragment) fail to ensure the appropriate coupling strength, whereby at leastone reaction fails to occur optimally), the detection and subsequent repairmentof that fragment (restoring back the adequate coupling strength between con-centrations of consecutive species of macromolecules) - take place without any3ppreciable passage of time. The second case, of course, does not discuss suchidealistic scenario; instead, it attempts to model the case when damage to a partof a pathway is detected after an appreciable time-lag, whereby the repairingmechanism starts to work only after an appreciable passage of time.Damage to a biochemical pathway though possible(as argued beforehand) areassumed to be not entirely common. Assuming that only rarely and accidentallydoes the synchronization among concentrations of interacting macromolecularspecies fail to satisfy the required optimality (and therefore cause the dam-age to the pathway), occurrences of such sub-optimal synchronization in anypart of a biochemical pathway ( P ) are assumed to take place as a Poisson pro-cess, characterized by an elementary flow with intensity λ . For the first case,the lack of synchronization in any part of P is detected immediately and theprocess of repairement of it starts without a delay. Using similar logic as theaforementioned one, we assume that the time required to repair the damagedsub-pathway can be described with a distribution of exponential nature with aparameter µ , whence the recovery process can be described as : f ( t ) = µe − µt ( t > . (1) Case -1):
For the first case, the repairing process starts as soon as the detection of thedamage takes place, which in turn, is assumed to take place immediately as thedamage takes place. Hence, the variable t (viz. time) in eqn-1 begins from thepoint of detection of damage, which implies that receovery process is describedstrictly in time range ( t > . Before this, viz. at the initial moment ( t = 0) , thebiochemical pathway ( P ) is assumed to be functioning without problem. Weattempt to find at first, the probability that at any arbitrarily chosen instance t,the pathway P is functioning properly and then, the probability that during anyarbitrarily chosen time interval (0 , t ) , P falters from its optimal functionaing atleast once; before attempting to evaluate the limiting probabilities of the statesof P .We denote the states of P as, ( s ) , when it is functioning properly and ( s ) ,when at least one part of it is malfunctioning and is being repaired; correspond-ingly, the probabilities p and p are assigned respectively.The Chapman-Kolmogorov equations (Sigman, hypertext link; Weisstein, hy-pertext link) for these states, viz. ( p ( t )) and ( p ( t )) can be constructed as : dp dt = µp − λp (2)and dp dt = λp − µp (3)4owever, since p + p = 1 , the redundancy in description can be eliminatedand by substituting p = 1 − p in eq n -2, we describe P with respect to p as : dp dt = µ − ( λ + µ ) p . (4)Solving eq n -4 for the initial condition p (0) = 1 , we obtain : p ( t ) = µλ + µ (cid:20) λµ e − ( λ + µ ) t (cid:21) (5)and therefore, p ( t ) = λλ + µ h − e − ( λ + µ ) t i (6)To solve the next part of our query that is to find the probability p ∗ ( t ) thatduring any arbitrarily chosen time interval (0 , t ) at least one part of P malfunc-tions at least once, we describe P with a new set of states; viz. ( s ) : when P never fails, and ( s ) : when at least one part of P malfunctions at least once.Here, solving the Chapman-Kolmogorov equation dp dt = − λp ∗ for the initialcondition p ∗ (0) = 1 , we get p ∗ ( t ) = e − λt we arrive at the probability that dur-ing the time interval (0 , t ) P malfunctions at least once; which is given by : p ∗ ( t ) = 1 − p ∗ ( t ) = 1 − e − λt .To find the limiting probabilities, we study eq n -5 and eq n -6 when t → ∞ ;whereby we arrive at the limiting probabilities of the states, given by : p = µλ + µ and p = λλ + µ . Case -2):