About Rate Constants in Cellucidate

Whereas the patterns in a rule determine what reactant species the rule applies to, the rate constant for a rule determines the rule's activity - on average how long it takes for the rule to be applied to the reactant species to produce the product species.  Thus, rate constants are vital determinants of the dynamic behavior of a model, but do not affect static analysis of a model (such as the influence map and contact map).

Cellucidate calculates rule activities for all rules using elementary chemical kinetics.  Thus, the allowable units for each rate constant are restricted based on the type of rule. For example, rate constants for first order rules (i.e., rules with a single connected pattern on the reactant side) must have units of [time^-1], and rate constants for second order rules (i.e., rules with two separate patterns on the reactant side) must have units of [concentration^-1 × time^-1].  For each rule, Cellucidate automatically chooses a list of relevant units based on the molecularity (order) of the rule.  It is up to the user to select relevant rate constant values and choose a set of units from the allowed list.

Cellucidate currently supports rate constants with units of concentration in molar (M), micromolar (μM or uM), nanomolar (nM), and molecules per reaction volume (molec), and units of time in seconds (s) and minutes (min).  When rules are assembled in a Kappa file for a stochastic or deterministic simulation, the rate constants are converted (as necessary) to concentration units in molecules per reaction volume (molec).  This conversion involves the reaction volume (in units of liters) and Avogadro's number.  For example, a rate constant (k) for a second order rule with the units of [nM^-1 × s^-1] is converted to units of [molec^-1 × s^-1] by the following formula: (k × 10⁹ nM/M) / (Volume × Avogadros_number).  Conversion to units of molec is required for stochastic simulation, and is done for deterministic simulation in Cellucidate merely for consistency.

It is often more advantageous to specify rate constants in units of concentration in molar, micromolar or nanomolar rather than units of concentration in molecules per reaction volume (molec) because rate constants in molec are not intrinsic properties of the reaction, but rather a combination of the intrinsic reaction rate and the volume in which the reaction is occurring.  For example, if we consider the rate constant for the binding of two proteins in different cell types of vastly different volumes (for example, human erythrocytes and  HeLa cells), the rate constant in units of [nM^-1 × s^-1] can be applied to both cell types, whereas the rate constant in units of [molec^-1 × s^-1] would be different in each cell type.  Additionally, rate constants are generally measured in with concentration units in molar (or micromolar, nanomolar, etc.), making this the more natural form for describing rate constants.

Typical Values for Rate Constants

The lack of reliable rate constants is one of the largest challenges to biological modeling.  For this reason, researchers often have to roughly estimate the values for rate constants when constructing models.  Below is a rough guide to typical rate constants for various types of rules or reactions.

Zeroth order reactions

Constitutive protein synthesis is a commonly modeled zeroth order reaction.  Typical rate constants are best derived from typical protein degradation rate constants (below) and the desired steady state amount of the protein: k_synth = k_deg × Steady_State_concentration. 

First order reactions

Protein degradation can be modeled as a first order process.  Half-lives of proteins can vary from a couple of minutes for rapidly degraded proteins to hours for very stable proteins.  This corresponds to degradation rate constants of 2E-2 s^-1 for unstable proteins to 2E-5 s^-1 for very stable proteins.

Dissociation of protein-protein complexes is probably the most commonly modeled first order reaction.  Typical dissociation rate constants for protein-protein complexes are between 0.1 s^-1 and 0.001 s^-1, wheras very unstable complexes may dissociate with a rate constant of 1 s^-1 and very stable complexes may dissociate with a rate constant of 0.0001 s^-1.

Another common first order reaction is protein phosphorylation or modification.  Typical phosphorylation rate constants for a kinase phosphorylating a bound substrate protein are in the range of 0.1 s^-1 to 10 s^-1 for active forms of kinases, and 0.01 s^-1 or lower for inactive kinases (for example, unphosphorylated MAPKs).

Second order reactions

Protein-protein association reactions are the most common second order reaction.  Typical rate constants for protein-protein interaction are on the order of 10⁵ M^-1 × s^-1, and for small molecule-protein interaction typical rate constants are on the order of 10⁶ M^-1 × s^-1.  The diffusion limit for protein-protein interactions is between 10⁸ M^-1 × s^-1 and 10¹⁰ M^-1 × s^-1, making this an absolute upper limit on physically realistic association rate constants.