Enhanced and non-monotonic effective kinetics of solute pulses under michaelis–Menten reactions

Abstract

Michaelis–Menten kinetics describe a broad range of physical, chemical, and biological processes. Since they are non-linear, spatial averaging of reaction kinetics is non-trivial, and it is not known how concentration gradients affect the global effective kinetics. Here, we use numerical simulations and theoretical developments to investigate the effective kinetics of diffusing solute pulses locally subject to Michaelis–Menten reaction kinetics. We find that coupled diffusion and reaction lead to non-monotonic effective kinetics that differ significantly from the local kinetics. The resulting effective reaction rates can be significantly enhanced compared to those of homogeneous batch reactors. We uncover the different regimes of effective kinetics as a function of the Damköhler number and Michaelis–Menten parameters and derive a theory that explains and quantifies these upscaled kinetics using a weakly-coupled description of reaction and diffusion. We illustrate the consequences of these findings on the accelerated consumption of nutrient pulses by bacteria. These results are relevant to a large spectrum of reactive systems characterized by heterogeneous concentration landscapes.

Introduction

Michaelis–Menten kinetics (Michaelis and Menten, 1913) occur in many natural and engineered reactive systems. They were originally developed as a model of catalytic reactions, where the reaction of interest is mediated by binding to a catalyst, leading to saturation effects (Michaelis, Menten, 1913, Segel, Slemrod, 1989). This type of kinetics has found applicability in a variety of contexts, such as microbial growth (Holmberg, 1982, Thullner, Regnier, 2019), chemotaxis (Novick-Cohen and Segel, 1984), solute transport in biological tissues (Ward, King, 2003, Hiltmann, Lory, 1983, McElwain, 1978, Nicholson, 1995), enzyme reactions (Horvath and Engasser, 1974), predator-prey models (Hsu et al., 2001), and reaction-diffusion in electrodes (Michael et al., 1996). In the context of bacterial growth, it is also known as Monod kinetics (Monod, 1949). They have been used extensively to model biodegradation of contaminants in hydrological and groundwater systems (Tompson, Knapp, Hanna, Taylor, 1994, Porta, la Cecilia, Guadagnini, Maggi, 2018, Barry, Prommer, Miller, Engesgaard, Brun, Zheng, 2002, Kindred, Celia, 1989, Blum, Hunkeler, Weede, Beyer, Grathwohl, Morasch, 2009, Ginn, Simmons, Wood, 1995). These kinetics display a simple non-linearity: the reaction rate is proportional to concentration at low concentrations and saturates to a constant above a threshold concentration. Analytical solutions exist for the Michaelis-Menten kinetics in batch conditions (Schnell, Mendoza, 1997, Maggi, la Cecilia, 2016). For non-homogeneous systems, the reaction-diffusion equation with Michaelis-Menten kinetics has been analyzed mathematically for different applications, leading to approximate solutions in some regimes (Ward, King, 2003, Hiltmann, Lory, 1983, McElwain, 1978, Shanthi, Ananthaswamy, Rajendran, 2013, Hsu, Waltman, 1993, Anderson, Arthurs, 1985, Anderson, Arthurs, 1980, Park, Agmon, 2008). Here we analyze the effect of chemical gradients on the average kinetic laws for local Michaelis-Menten kinetics. We investigate whether non-homogeneities in concentrations may lead to enhanced or reduced average reaction rates compared with batch kinetics, characterized by homogeneous concentrations.

Under non-linear kinetics, unresolved concentration gradients lead to effective macroscopic reactive transport laws that are different from microscopic laws (Battiato, Tartakovsky, 2011, Battiato, Tartakovsky, Tartakovsky, Scheibe, 2009, Meile, Tuncay, 2006, Guo, Quintard, Laouafa, 2015). In the context of Michaelis-Menten reactions, the effect of mass transfer limitations on effective macroscopic kinetics has been studied with an emphasis on bioavailability limitations when micro-organisms are located on solid surfaces (Heße, Radu, Thullner, Attinger, 2009, Wood, Radakovich, Golfier, 2007, Hesse, Harms, Attinger, Thullner, 2010) or more generally distributed in space (Schmidt et al., 2018). Mixing limitation with Michaelis–Menten kinetics have also been investigated in the context of reactive fronts, where reactants are spatially segregated and mixing is the limiting step to bring reactants into contact (Sole-Mari, Fernàndez-Garcia, Rodríguez-Escales, Sanchez-Vila, 2017, Ding, Benson, 2015). Here we study situations where nutrients or reactants are released as discrete pulses in time and space, which encompasses a large spectrum of natural and engineered systems. Examples include pulse of nutrients in soil (König, Worrich, Banitz, Centler, Harms, Kästner, Miltner, Wick, Thullner, Frank, 2018, Waring, Powers, 2016), plants (Orcutt and Nilsen, 2000), aquifers (Bochet et al., 2020) or catchments (Weigelhofer et al., 2018), which are often consumed by biological agents through Michaelis–Menten kinetics (Haefner, 2005). While other types of non-homogeneous initial conditions could be considered, we argue that the general impact of concentration gradients on the average kinetics will be similar as for pulses.

We study the effective kinetics of diffusing pulses of a single chemical species undergoing degradation with Michaelis–Menten kinetics. We assume that the local kinetics are uniform in space and hence focus on the effect of spatial and temporal changes in reactant concentration on the effective kinetics. We approximate these nonlinear kinetics by a sharp crossover from a linear dependency of the degradation rate on c for concentrations lower than the crossover concentration, to a saturated, constant rate above it. We investigate the dependency of the effective kinetics on the Damköhler number $\mathrm{Da}$ and the ratio α between the kinetics’ crossover concentration and the initial concentration. We develop a semi-analytical framework relying on a weak-coupling approximation regarding diffusion and reaction. The results compare favorably to numerical simulations of the coupled equations. Fully-analytical descriptions are also derived for asymptotic regimes corresponding respectively to reaction- and diffusion-dominated dynamics.

In the following, we first present, in Section 2, a mathematical description of the dynamics, including the solution under well-mixed conditions, which will serve as the reference scenario. Next, Section 3 is concerned with analysing the dynamics of the effective reaction rate as a function of the Damköhler number and α based on numerical simulations. Section 4 is devoted to the derivation of the semi-analytical theory relying on the approximation of weakly-coupled diffusion and reaction. Section 5 explores the consequences of our results in the context of the consumption of nutrients by bacteria. Conclusions are drawn and the results discussed in terms of their relevance to natural systems in Section 6. Additional technical derivations regarding the analytical theory and details on the performance of the weakly coupled approximation may be found in appendix.