Basal Expression Level
The basal expression of mRNA, or a protein, refers to its “default” expression level.
Essentially, it’s the level of mRNA/ protein that will be constitutively produced in a cell independent
of the activity of its promoter.
Cellular automata are discrete models, that were introduced in the 1940s, that act as a model of computation.
They model the state of a cell at a particular point in time. Cells are represented on a grid, and their state
(on/ off/, empty/alive) changes according to a set of rules which is dependent on the state of their
Game of Life
The Game of Life, (which we adapted for our purposes), relates a particular case of a
2D cellular automata that is known to show complex behaviors.
The Game of Life was developed in the 1970s by John Conway. Cells which can be either in the "born"/"alive" or "dead" state are initialized on a 2D grid. The idea is to visually represent the evolution
of the state of the cells over time according to a set of rules.
The Hill equation is used in biology to describe the interaction between a ligand and its binding site. The general form of the Hill equation is expressed as:
f(L) = Fraction of the protein concentration bound to the ligand
L = free ligand concentration
n = Hill coefficient
Kd = Dissociation constant which follows mass action kinetics
The Hill equation is often adapted in mathematical modelling to allow for a more flexible modelling of the dynamics of a promoter.
In our case we implemented a form of the Hill equation that had been adapted to represent a photo-activated cellular mechanism to determine the rate of photo-activation (k1) of our transcription factor.
Two parameters were added to the equation to account for our modelling needs:
a = accounts for the basal expression level of the promoter (LuxI)
k = accounts for the maximum expression of the promoter due to light intensity 0
Mass Action Kinetics
Mass action kinetics describe chemical reactions. It assumes that the rate of a chemical reaction is proportional to the concentration of the initial reactants.
Mass action kinetics assumes:
Given an Ordinary Differential Equation (ODE):
Where k is known as the rate constant, its units depend on the order of the reaction, i.e. the number of reactants.
The Michaelis Menten equation was created to describe the rate kinetics in enzymatic reactions. Computational biologists commonly use the Michaelis Menten equation to describe the rate kinetics of cellular mechanisms. The steps described by the Michaelis Menten kinetics, can be represented schematically
in the following equation:
E = enzyme
S = substrate
ES = enzyme substrate complex
P = product
The rate at which the final product is formed is given by the following equation:
v = rate of product concentration over time
P = concentration of product
Vmax = maximal rate of reaction
S = concentration of the substrate
Km = Michaelis Menten constant (value of concentration of ‘the substrate when the velocity = ½ maximum velocity)
We assume the system is at steady state, i.e. the concentration of the [ES] complex remains constant. (It can be assumed
to be constant as it represents a species that is constitutively expressed by a cell)
The Planck-Einstein relationship (quantum mechanics) is the formula that relates energy given by a photon to frequency.
E = energy (J)
h = Planck's Constant (Js)
f = Frequency (Hz)
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