About Singulator

Recent computational methodologies, such as agent-based modelling, support the search for explanatory insight into the collective behaviour of molecules and bring forward the discussion of intrinsic noise while simulating biological events. The present work contributes to this line of research by assembling and validating a molecular-scale agent-based model for the simulation of enzymatic reactions at experimentally measured concentrations. The model incorporates stochasticity and spatial dependence, using diffusing and reacting particles with physical dimensions. We started by adjusting the information from classical theories, namely enzymatic rates and diffusion coefficients, to the information required by the computational agents, i.e. collision efficiency, interaction logic between agents, the time scale associated with interactions (e.g. kinetics), and agent velocity. We developed specific strategies to implement the model in a way that could use and reproduce experimental measures of diffusion and kinetics, in the form of Michaelis-Menten parameters. The velocity of the agents was defined in an iterative manner to match the corresponding theoretical diffusion coefficients.

Agents capable of diffusion and reaction take on physical dimensions based on the data reported by scientific literature and the BRENDA database. Enzymes and metabolites are represented by agents with spherical approximations (by the hydrodynamic or van der Walls radius) of the excluded volume of the biomolecules. The diffusion coefficients were calculated using the Stokes-Einstein equation for diffusion of spherical particles in a liquid.

Singulator workflow

Enzymatic reactions are defined by the computational parameters simkcat, simKm and reaction radius. The simkcat is the number of time steps between the formation of enzyme-substrate reaction complex and the product release. The simKm quantifies the probability of a successful collision between an interacting enzyme and substrate. And, the reaction radius enables the enzyme agent to detect substrate agents off the immediate vicinity and test more possible interactions.

Diffusion-controlled enzymes

Initially, we performed a simulation where the transformation of substrate into product would take only one time step to occur, which is equivalent in our system to the minimum value of kcat. This implies that, under conditions of substrate saturation, the simulated reaction should be limited by diffusion.

In a first approach, as simkcat increases, the rate of formation of product agents during the time steps becomes increasingly linear .This is to be expected of reactions limited by the catalytic step.

virtual enzymatic

The relative occupancy of enzymes, defined as the percentage of the total number of enzymes that are bound in the enzyme-substrate complex (ES/Et %), was also used to distinguish simulations where the enzymatic reaction was limited by diffusion, as opposed to being limited by the product release step. That is, systems with higher simkcat are constant at near 100% occupancy of enzymes Fig (B).

The matching ratios of simkcat and kcat are also useful to evaluate diffusion control and the coherency of simulation behaviour.

virtual enzymatic

Calibration of Km

The calibration of simKm to Km was based on the reproduction of experimental assays of kinetic parameters, which measure the velocity of the reaction for different concentrations of substrate, below the substrate saturation level.

Strategies were tested using data of two real isomerases: the 2-hydroxymuconate tautomerase from Pseudomonas putida (EC, UniProt ID Q01468) and the 2-hydroxymuconate tautomerase was applied to a different isomerase, the Steroid Delta-isomerase, also from Pseudomonas putida (EC, UniProt ID P07445).

Simulation results that are in the same order of magnitude of the values of Michaelis-Menten parameters.

Km calibration

Assessing Intrinsic Noise

The low concentrations of some of the key enzymes and metabolites inside the cell can make local substrate fluctuations important sources of cellular variability. While at high concentrations replicate simulations have practically the same exact behaviour, with very few local discrepancies of the moment of catalytic turnover, at lower concentrations the turnover events are more dispersed across time steps (Fig A).

Additionally, the average relative deviation of the number of product agents was calculated to assess the impact that the initial substrate concentration has on product formation, both for the 2-hydroxymuconate tautomerase enzyme (Fig B) and steroid-delta isomerase enzyme (Fig C).

Km calibration


Ultimately, the goal of molecular-scale computational models of cellular environments is to grow our understanding about the differences between biomolecular behaviour observed in vitro and in vivo. The simulations and calculations described here represent the first attempt to build such a model using diffusing and reacting particles with realistic physical dimensions, and incorporating stochasticity and spatial dependence in a three-dimensional environment. This experiment exposes the computational requirements imposed by a realistic scenario and raises discussion about future lines of research and development for agent-based biomodelling.

Contact us

nazevedo@fe.up.pt Nuno Azevedo Project Leader
analia@uvigo.es Anália Lourenço Project Leader
riverola@uvigo.es Florentino Fdez-Riverola Project Leader
mpperez3@esei.uvigo.es Martín Pérez Pérez Since 2013
gprodriguez2@esei.uvigo.es Gael Pérez Rodríguez Since 2013
gdbmonteiro@gmail.com Gonçalo Monteiro Since 2014
denise.neves.gameiro@gmail.com Denise Neves Since 2014