Team:British Columbia/Modelling/Adsorptions
From 2014.igem.org
Adsorptions
 Model for caulobacter Attachment to Chalcopyrite

The mechanism for caulobacter (~1um diameter) attachment to particles of chalcopyrite (~100um) share many similarities with a standard adsorption problem. The Langmuir isotherm has been adopted for modelling caulobacter to chalcopyrite. The model defines 3 distinct regions, the surface, subsurface and bulk. The surface is the interface between chalcopyrite and the medium, with the medium being the suspension fluid holding both the caulobactor and chalcopyrite. The surface consist of a finite number of sites for caulobactor attachment. The subsurface is an arbitrary volume around the surface where we allow the concentration of caulobactor to fluctuate. The bulk is everything else, it differs from the subsurface as we assume the concentration of caulobactor is relatively constant. This model hence assumes the total number of caulobacter in the system is much greater than the totally number of attachment sites on chalcopyrite.
The 5 parameters are.
$$k_1$$  rate constant associated with caulobacter attachment from chalcopyrite [1/time] $$k_2$$  rate constant associated with caulobacter detachment from chalcopyrite [1/time] $$k_3$$  rate constant associated with caulobacter mass flow between subsurface and bulk [1/time] $$\widehat{\Gamma}$$  Concentration of all attachment sites within the subsurface volume [number of sites/unit volume] $$C^{\infty}$$  Concentration of caulobacter in bulk [number of caulobacter/unit volume]
The 2 time dependant variables are.$$C$$  Concentration of caulobacter in subsurface [number of cells/unit volume] $$\Gamma$$  Concentration of attached caulobacter in subsurface [number of caulobacter/unit volume] Mass flow from bulk to subsurface is given by $$k_3(C^{\infty}C(t)))$$
Attachment rate is given by $$k_{1}C(t)\left [ 1\frac{\Gamma(t)}{\widehat{\Gamma}} \right ]$$
Detachment rate is given by $$k_{2}\Gamma$$
The ODE system is given by
$$\frac{d}{dt}\Gamma(t)=k_{1}C(t)\left [ 1\frac{\Gamma(t)}{\widehat{\Gamma}} \right ]$$ $$\frac{d}{dt}C(t)=\frac{d}{dt}\Gamma(t)+k_3(C^{\infty}C(t)))$$The total number of free sites can be estimated given the surface area of chalcopyrite particles and foot print of caulobacter. A matlab code was written to randomly place circles (simulating the caulobacter) of 1 unit diameter around a square. The x,y coordinates of the circles were generated with a uniform random distribution, each time coordinate was generated it was checked against other circles that have already been placed to see if they collide before placing onto the square. The script stops when 1000 consecutive attempted placements fail. The script was run for a surface 100 squared unit 5 times and verified against a surface of 1000 squared units run twice.
As the results for 100 and 1000 unit area surface both give reasonably close values for circles per unit area it is safe to assume there are about 0.55 sites for caulobacter attachment per square um of chalcopyrite.
As the results for 100 and 1000 units area both give reasonably close values for circles per unit area it can be assumed there are about 0.55 sites for caulobacter attachment per square um of chalcopyrite. A sped up visual of the matlab code running is provided below to show the circles being generated and randomly placed onto the surface.
The following test plot consist of empirical data shows the effect of varying caulobacter concentration in the bulk. The graph shows that increasing the bulk concentration increases both the initial rate of adsorption and the equilibrium concentration of attached caulobacter.
The following test plot consist of empirical data shows the effect of varying diffusion rate between bulk and subsurface. The graph shows that increasing the diffusion rate increases both the initial rate of adsorption but the equilibrium concentration of attached caulobacter remains constant.