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A linear biochemical pathway is a chain of enzyme-catalyzed reaction steps where the product of one reaction becomes the substrate for the next reaction. The molecules progress through the pathway sequentially from the starting substrate to the final product. Each step in the pathway is usually facilitated by a different specific enzyme that catalyzes the chemical transformation. An example includes DNA replication, which connects the starting substrate and the end product in a straightforward sequence.
Biological cells consume nutrients to sustain life. These nutrients are broken down to smaller molecules. Some of the molecules are used in the cells for various biological functions, and others are reassembled into more complex structures required for life. The breakdown and reassembly of nutrients is called metabolism. An individual cell contains thousands of different kinds of small molecules, such as sugars, lipids, and amino acids. The interconversion of these molecules is carried out by catalysts called enzymes. For example, the most widely studied bacterium, E. coli strain K-12, is able to produce about 2,338 metabolic enzymes. [1] These enzymes collectively form a complex web of reactions comprising pathways by which substrates (including nutients and intermediates) are converted to products (other intermediates and end-products).
The figure below shows a four step pathway, with intermediates, and . To sustain a steady-state, the boundary species and are fixed. Each step is catalyzed by an enzyme, .
Linear pathways follow a step-by-step sequence, where each enzymatic reaction results in the transformation of a substrate into an intermediate product. This intermediate is processed by subsequent enzymes until the final product is synthesized.
A linear pathway can be studied in various ways. Multiple computer simulations can be run to try to understand the pathway's behavior. Another way to understand the properties of a linear pathway is to take a more analytical approach. Analytical solutions can be derived for the steady-state if simple mass-action kinetics are assumed. [2] [3] [4] Analytical solutions for the steady-state when assuming Michaelis-Menten kinetics can be obtained [5] [6] but are quite often avoided. Instead, such models are linearized. The three approaches that are usually used are therefore:
It is possible to build a computer simulation of a linear biochemical pathway. This can be done by building a simple model that describes each intermediate through a differential equation. The differential equations can be written by invoking mass conservation. For example, for the linear pathway:
where and are fixed boundary species, the non-fixed intermediate can be described using the differential equation:
The rate of change of the non-fixed intermediates and can be written in the same way:
To run a simulation the rates, need to be defined. If mass-action kinetics are assumed for the reaction rates, then the differential equation can be written as:
If values are assigned to the rate constants, , and the fixed species and the differential equations can be solved.
Computer simulations can only yield so much insight, as one would be required to run simulations on a wide range of parameter values, which can be unwieldy. A generally more powerful way to understand the properties of a model is to solve the differential equations analytically.
Analytical solutions are possible if simple mass-action kinetics on each reaction step are assumed:
where and are the forward and reverse rate-constants, respectively. is the substrate and the product. If the equilibrium constant for this reaction is:
The mass-action kinetic equation can be modified to be:
Given the reaction rates, the differential equations describing the rates of change of the species can be described. For example, the rate of change of will equal:
By setting the differential equations to zero, the steady-state concentration for the species can be derived. From here, the pathway flux equation can be determined. For the three-step pathway, the steady-state concentrations of and are given by:
Inserting either or into one of the rate laws will give the steady-state pathway flux, :
A pattern can be seen in this equation such that, in general, for a linear pathway of steps, the steady-state pathway flux is given by:
Note that the pathway flux is a function of all the kinetic and thermodynamic parameters. This means there is no single parameter that determines the flux completely. If is equated to enzyme activity, then every enzyme in the pathway has some influence over the flux.
Given the flux expression, it is possible to derive the flux control coefficients by differentiation and scaling of the flux expression. This can be done for the general case of steps:
This result yields two corollaries:
For the three-step linear chain, the flux control coefficients are given by:
where is given by:
Given these results, there are some patterns:
With more moderate equilibrium constants, perturbations can travel upstream as well as downstream. For example, a perturbation at the last step, , is better able to influence the reaction rates upstream, which results in an alteration in the steady-state flux.
An important result can be obtained if all are set as equal to each other. Under these conditions, the flux control coefficient is proportional to the numerator. That is:
If it is assumed that the equilibrium constants are all greater than 1.0, as earlier steps have more terms, it must mean that earlier steps will, in general, have high larger flux control coefficients. In a linear chain of reaction steps, flux control will tend to be biased towards the front of the pathway. From a metabolic engineering or drug-targeting perspective, preference should be given to targeting the earlier steps in a pathway since they have the greatest effect on pathway flux. Note that this rule only applies to pathways without negative feedback loops. [7]
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.
In chemistry, the rate equation is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters only. For many reactions, the initial rate is given by a power law such as
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Used in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar to that used in heat transfer to describe the flow of heat in a solid. The steady-state flow of groundwater is described by a form of the Laplace equation, which is a form of potential flow and has analogs in numerous fields.
The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes. It states that at equilibrium, each elementary process is in equilibrium with its reverse process.
Enzyme kinetics is the study of the rates of enzyme-catalysed chemical reactions. In enzyme kinetics, the reaction rate is measured and the effects of varying the conditions of the reaction are investigated. Studying an enzyme's kinetics in this way can reveal the catalytic mechanism of this enzyme, its role in metabolism, how its activity is controlled, and how a drug or a modifier might affect the rate.
Flux balance analysis (FBA) is a mathematical method for simulating metabolism in genome-scale reconstructions of metabolic networks. In comparison to traditional methods of modeling, FBA is less intensive in terms of the input data required for constructing the model. Simulations performed using FBA are computationally inexpensive and can calculate steady-state metabolic fluxes for large models in a few seconds on modern personal computers. The related method of metabolic pathway analysis seeks to find and list all possible pathways between metabolites.
Metabolic control analysis (MCA) is a mathematical framework for describing metabolic, signaling, and genetic pathways. MCA quantifies how variables, such as fluxes and species concentrations, depend on network parameters. In particular, it is able to describe how network-dependent properties, called control coefficients, depend on local properties called elasticities or Elasticity Coefficients.
The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift–diffusion equation, or (generic) scalar transport equation.
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Transient kinetic isotope effects occur when the reaction leading to isotope fractionation does not follow pure first-order kinetics (FOK) and therefore isotopic effects cannot be described with the classical equilibrium fractionation equations or with steady-state kinetic fractionation equations. In these instances, the general equations for biochemical isotope kinetics (GEBIK) and the general equations for biochemical isotope fractionation (GEBIF) can be used.
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In chemistry, control coefficients are used to describe how much influence a given reaction step has on the steady-state flux or species concentration level. In practice, this can be accomplished by changing the expression level of a given enzyme and measuring the resulting changes in flux and metabolite levels. Control coefficients form a central component of metabolic control analysis.
In biochemistry, a rate-limiting step is a step that controls the rate of a series of biochemical reactions. The statement is, however, a misunderstanding of how a sequence of enzyme catalyzed reaction steps operate. Rather than a single step controlling the rate, it has been discovered that multiple steps control the rate. Moreover, each controlling step controls the rate to varying degrees.
Control coefficients measure the response of a biochemical pathway to changes in enzyme activity. The response coefficient, as originally defined by Kacser and Burns, is a measure of how external factors such as inhibitors, pharmaceutical drugs, or boundary species affect the steady-state fluxes and species concentrations. The flux response coefficient is defined by:
In metabolic control analysis, a variety of theorems have been discovered and discussed in the literature. The most well known of these are flux and concentration control coefficient summation relationships. These theorems are the result of the stoichiometric structure and mass conservation properties of biochemical networks. Equivalent theorems have not been found, for example, in electrical or economic systems.
The stoichiometric structure and mass-conservation properties of biochemical pathways gives rise to a series of theorems or relationships between the control coefficients and the control coefficients and elasticities. There are a large number of such relationships depending on the pathway configuration which have been documented and discovered by various authors. The term theorem has been used to describe these relationships because they can be proved in terms of more elementary concepts. The operational proofs in particular are of this nature.
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