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MBW:Governing Fluid Dynamics of Non-Motile Bacteria in Varying Gravitational Domains

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Executive Summary

It is difficult in space to conduct a truly controlled biological experiment. Many properties of space flight, specifically micro-gravity, have yet unknown effects on organisms. Variations can be caused to the interior of a cell, the microenvironment directly surrounding the cell, and the macro-scale containing-medium. Conducting a successful experiment requires strict regulation and understanding of all variables in effect and parameters used in controlling the system. For a biological system, effects can be summarized as either direct (stress, strain, weight and shear force) or indirect (altered bulk fluid convection). The goal of this paper is to discuss the fluid dynamics of a bacterial environment as first discussed in a paper by Clauss et. All (2004)[1]. Analyzing the dynamics of biological phenomena and the equipment used to observe them will hopefully lead to an increased skill in experimental design. Mainly the effect of micro-g on the chemical concentrations in boundary fluid layer surrounding a cell will be discussed.

Introduction

History

Beginning with the first manned space flights, biologists and astronauts alike wondered about the change in environment and the resulting effects to a human body, a plant, or a mere microorganism. Many cause-and-effect pathways have been hypothesized and researched including increased radiation levels, hyper-g during launch, random-g (vibration) and obviously micro-g. Micro-gravity is defined as gravity with magnitude that is more than two levels of magnitude below 1-g baseline. Typically gravity in orbit has a magnitude of E-4 g [2]. This paper will concentrate explicitly on single celled organisms, such a simple bacteria or protozoa; and how their behavior is altered by micro-g scenarios.

Data has been collected that shows a few common trends in bacterial growth. First, it has been noted that bacterial cells, after being exposed to micro-g, become largely more virulent. This can be detrimental both to astronauts in space, as well as the populations on earth who are exposed to the stronger bacterial strains upon re-entry. It has also been noted that cellular metabolism increases in efficiency. That is, given a specific number of cells, and certain amount of substrate, more by-products will be produced in space than on earth, and the cells will grow larger. This seems counter-intuitive, because as we will later discuss, mass-transport (and therefore the supply of fuel) around cells decreases in micro-g. However, the phenomena can be explained similarly to the effect of batch-feeding a culture of cells: When you feed a solution of cells just enough to survive, and no more, the cells use the substrate to the highest efficiency. Whereas, when cells are over-fed, they have no stimulus to work efficiently.

Most effects of micro-g are characterized in comparison to the 1-g equivalent system. They can either be observed in a true micro-g environment, or some simulation there-of (such as a clinostat or rotating wall vessel). All effects are often grouped into two categories: direct and indirect. Direct effects result from changes in acceleration, internal stress, deformation, or weight, while indirection effects arise passively from external system changes. Observations can also be effected heavily by the measurement method, the procedural discourse, and the specific equipment used.

Direct Gravitational Effects

Direct effects of gravity classically simplify into deformation (weight) and displacement (motion). For a biological system, these effects can be calculated given some information about the system such as mass, density, geometry, supporting framework, and other material properties. For a given inertial environment (micro-g, 1-g, hyper-g, random-g) the position, density and internal attachment of all the components of an organism and be used to calculate the perceived weight and center of mass. Using these characteristics, other state values such as stress and strain can be calculated.

Beyond static forces, an object can also feel external forces due to such effects as fluid convection. If the containing fluids are viscous and have different densities, gravity driven convection currents will form. Also, if the object itself is of different density than the surrounding environment, it will rise or fall. Then, the object will experience a pressure force on one side due to ‘pushing’ through the fluid it is contained in. Shear forces can also cause structural damage to the cell membrane, or integral membrane proteins and receptors. The damage due to shear force is particularly relevant when comparing maximal cell size in 1-g and micro-g. As gravity decreases in magnitude, convection due to differing densities also decreases. This causes a decrease in shear forces on the cell walls, allowing cells to grow larger than at 1-g, without the hazard of rupturing.

Indirect Gravitational Effects

Indirect effects of micro-g are often harder to identify and measure than direct effects. An indirect effect occurs due to some change in the environment of the cells. This could be changing solute concentration levels, fluid flow rates or densities or altered cell distributions. Most indirect effects of gravity can be contributed to differing behavior of mass transport. Simple diffusion is dependent only on concentration levels, kinetic energy of the fluid, and diffusion constants; so will not be affected by changing gravity. However, diffusion is just one way in which a cell receives nutrients and expels waste. Other forms of mass transport such as convection currents no longer exist in micro-g. Convective currents arise due to differing densities (and therefore weights) of chemicals in heterogeneous solution. As weight is directly dependent on gravity, convection is essentially reduced to zero in space. With no convective flow, less solute is carried past the cell.

Procedure and Design

Much of the difficulty in interpreting data taken in space comes from the variability added when adapting an experiment for a micro-g laboratory setting. Often the norms and practices, which are commonplace throughout all labs, are hard to adhere to strictly in space. Simple processes such as measuring volume, plating agar or centrifuging a sample become extremely hard to perform. The normal vibrations and air movement of a ground lab are not present in a space lab, instead replaced by artificially circulated and filtered air supplies, and unpredictable amounts of vibration.

Equipment

Often, scientists will try to simulate micro-g phenomena on earth to avoid the extravagant costs of sending personnel, supplies, and equipment into space. Techniques are used to generate a control result to compare actual results taken from space. The method is known as clinorotation. The general idea is form an environment with minimal shear forces, fluid flow and gravitational effects, as well as allowing the cells to distribute uniformly throughout the solution. Rotating Wall Vessels (RWV’s) or clinostats are generally utilized. A clinostat is a cylinder of solution oriented with the axis perpendicular to the gravity gradient. This horizontal cylinder is then slowly rotated causing the solution to rotate slowly as well. Because cells are constantly being reoriented, the gravity vector is therefore very close to random over a long time domain. The hope is that the net result is that the cell will experience ‘weightlessness’, where in-fact it is more accurately experiencing ‘random weight’. This approach prevents the cells from ever sedimenting; but currents are still present in this method, as the heavier fluids will still sink relative to the less-dense fluids [3].

Model

Model Developement

There are two key equations that dictate the mass transport in the fluid environment around a single cell. Equation 1 uses the Navier Stokes equation with the Boussinesq approximation to satisfy conservation of momentum, while accounting for the differing densities of substrate and bi-product (note the subscript i's). It should be noted that this model only considers the consumption of glucose and assumes the bi-product is solely acetate; produced at a rate of half the glucose consumed by the cell.

{\frac  {\partial {\vec  u}}{\partial t}}+{\vec  u}\cdot \bigtriangledown u=-{\frac  {1}{\rho _{f}}}\bigtriangledown \rho +v\bigtriangledown ^{2}{\vec  u}+{\vec  g}\beta _{i}\Delta C_{i}\quad \quad \quad \quad (1)


\beta =1/{{\Bigg (}\rho {\bigg (}{\frac  {\partial \rho }{\partial C}}{\bigg )}{\Bigg )}}\quad \quad \quad \quad (1.a)


\beta represents calculated solute expansion coefficients, which can be used in finding how much the cell metabolism changed the surrounding density .


Equation 2 is a general diffusion equation with the addition of the buoyancy driven velocity from equation 1. It satisfies mass conservation. All cellular chemical reactions occur within the cells interior, thus the flux of glucose and the by-products are modeled extracellulary by.

{\frac  {\partial C_{i}}{\partial t}}+{\vec  u}\cdot \bigtriangledown C_{i}=D_{i}\bigtriangledown ^{2}C_{i}\quad \quad \quad \quad (2)


With the concentrations known at specific times as well as radial locations it was possible to calculate the relative density change. This is needed when determining later the convective current due to changing densities. This was done using the following approximation, and the same beta's as before.


{\Delta \rho }\approx {\beta _{1}\Delta C_{1}+\beta _{2}\Delta C_{2}}\quad \quad \quad \quad (3)


Once the change in density was calculated it was possible to calculate the non-dimensional Peclet number. The Pecet relates the relevance of convective vs diffusive transport. It can be thought of as a ration of their effects. The smaller the Peclet number, the more diffusion-dominated the system is. A Peclet number smaller then .1 is thought to be dictated by diffusion. The larger the number the system is driven more by convection. A Peclet number greater then 10 is thought be dictated by convection. Numbers between .1 and 10 have cross coupled effects. A figure containing the relevance of single cell mass transport can be found in #Results and Discussion. The non-dimentionalazation of the Pe number can be seen below in equation 4.


Pe={\frac  {V_{{max,local}}L_{{cell}}}{D_{{glucose}}}}\quad \quad \quad \quad (4)


Here V_{{max}} is the maximum velocity resulting from convective terms. These can be related to more non-dimensional numbers, Gr the Grashof number and Sc the Schmidt number. L is a characteristic length and for our purposes is the diameter of the cell. v is the kinematic viscosity.


V_{{max,local}}={\frac  {v}{L}}{\sqrt  {{\Bigg (}{\frac  {Gr}{Sc}}{\Bigg )}}}\quad \quad \quad \quad (5)


The Grashof number relates the ratio of Buoyancy to Viscous forces \Delta \rho is the change of density calculated earlier.

where Gr={\Bigg (}{\frac  {gL^{3}(\Delta \rho /\rho )}{v^{2}}}{\Bigg )}\quad \quad \quad \quad (5.a)


The Schmidt number is a non-dimetional number that is defined by the ratio of viscosity to mass diffusivity.

Sc={\frac  {v}{D}}\quad \quad \quad \quad \quad \quad \quad \quad \quad (5.b)

The previous two equations describe characteristics of the medium the cells are suspended in. To learn more about equation 5.a and 5.b, one can look in a classic fluid dynamics textbook. But their derivation will not be included in the scope of this paper. The previous equations can be rearranged such that the Peclet number is as follows in equation 6. Pe={\sqrt  {{\frac  {g(\Delta \rho /\rho )L^{3}}{vD}}}}\quad \quad \quad \quad (6)

Parameters, Measured Values, and Boundary Conditions

We will assume that we have just began a batch feeding cycle for a solution of cells in suspension. Therefore the glucose concentration is known, and the acetate concentration is initially zero. We will also used previously calculated values for the density of a glucose solution at the given concentration. The fluid is fully stationary. Fig1.png


For constants, such as gravity, solute diffusion, fluid viscosity, and solute flux, we used values taken from the Benoit (2008) paper. They took values from previous research or experiments. Betas, can be seen in the table above. Fig2.png

Adapting Model for Flocculation

Thus far, we have assumed that bacterial cells are completely stationary and suspended in solution. It has also been assumed that the majority of the cells are spaced sufficiently that there is no interaction. Now, we will investigate the effect of cellular aggregation, or flocculation; that is, cells growing in clusters of varying size. We will assume the glucose consumption rate of an active cell remains constant, R_{{cell}}. The rate is merely the glucose flux (see table 2) multiplied by the surface area of the cell. These values have been measured in experiments. It is known that when flocs form, given certain conditions, a core of dead cells (inactive) will form[4]. Therefore, we will establish a maximum and minimum value for the consumption rate of the whole floc, which will allow us to find the glucose flux at the boundary of the cell. The maximum glucose consumption for the cell will be achieved when the entire biomass is composed of active cells. The minimum will be approximated as when only the outermost layer (shell) of cells in alive. In both cases, the total flux \Phi _{T} can be calculated as by dividing the total consumption rate, R_{T} by the total outer surface-area of the floc. This assumes the floc is strictly spherical, and of radius r_{T}.

\Phi _{T}={\frac  {R_{T}}{SA_{T}}}\approx {\frac  {nR_{{cell}}}{4\pi {r_{T}}^{2}}}\quad \quad \quad \quad (7)

This means we can approximate the maximal boundary glucose flux, \Phi _{{max}}, and the minimal flux, \Phi _{{min}}, if we can calculate the number of active cells, n, in each case.

n_{{active,max}}\approx {\frac  {V_{T}}{V_{{cell}}}}={{\Bigg (}{\frac  {r_{T}}{r_{{cell}}}}{\Bigg )}}^{3}\quad \quad \quad \quad (7.a)

n_{{active,min}}\approx {\frac  {V_{{shell}}}{V_{{cell}}}}={\frac  {{\Big (}{r_{T}}^{3}-{r_{{inactive}}}^{3}{\Big )}}{({r_{{cell}}})^{3}}}={\frac  {{\bigg (}{r_{T}}^{3}-{(r_{T}-2r_{{cell}})}^{3}{\bigg )}}{{(r_{{cell}})}^{3}}}\quad \quad \quad \quad (7.b)

With the newly calculated glucose flux values, we can recalculate all the characteristic values from the first section #Model Developement. The two flux values will give us bounds on where we the actual values are likely to fall. Our assumptions go to both extremes of floc dynamics, where real flocs fall somewhere in the middle.

Results and Discussion

Original Model

In the Benoit paper(2008), they used the same equations we did, but applied them to multiple spatial domains. They investigated the dynamics in a vertical cylinder, cuvette, as well as around an isolated, non-motile cell. We will compare our results with the results of their isolated cell section, as that was the sole dynamic we investigated. The Benoit group used a far more powerful solver software package than we had access to, however, we still ended up with very similar numbers. The values of maximal density change, \Delta \rho , V_{{max}}, and Pe_{{max}} agreed within roughly 25% error. The comparison of results from 1-g simulations are shown below.

Results.png


Below are three surface plots that best highlight the dynamics directly surrounding a cell; note the log-scale. First is a plot of glucose concentration at differing times and distances away from the cell. The initial condition is a typical glucose concentration for cell cultures. We used matlab's partial differential equation solver to create the solution curves. Glucose Concentration.jpg

Next (below) is a similar plot showing both the glucose concentration as well as the acetate (by-product) concentration. Glucose is consumed in the the cell, so levels decrease near the cell; while acetate is produced by the cell, and diffuses outward. The glucose concentration is much larger than the acetate concentration. Glucose and Acetate Contrentration.jpg

Finally, we plotted the Peclet Number varying in space and time. Note the magnitudes of Pe is in the range of (e-3), meaning that diffusion is the dominant phenomena for a single cell, even with gravity causing some convection. The force of convection is proportionally small enough, for a single cell of radius 1 micron, that it can be essentially ignored.

Peclet Number.jpg

Adding Flocculation

Knowing that our basic model was working well for a single cell, we adapted our analysis to include flocculation. We calculated an upper and lower bound for the surface glucose flux of a flocculate at a given size. Using these new values for flux, as well as cell radius (now floc radius) we were able to recalculate the Peclet number for flocs of given differing sizes. The plot of Pe varying with floc size is shown below.

Peclet Number for Flocs.jpg

The plot tells a lot about the dynamics of a floc. First, the average size of an E. Coli floc is between 20 um and 60 um [5]. In this range, the Peclet number rises significantly compared to a single cell. The magnitude increases into the (.1) to (10.0) range, which means that now both convection and diffusion are important factors is mass-transfer of solutes to and away from the cell. This is a slightly different result than was found in the Benoit paper. In their results, they claimed a cell would need to be roughly 100 um in radius before the peclet number rose enough to create a convection dominated environment. This is assuming the cell was singular, not a cluster of separate cells. However, in our simulation we found much smaller radiuses produced convective flow. This holds even when assuming minimal glucose consumption (and thus minimal Peclet number).

Below is a similar plot that was generated by the Benoit group. It shows how the peclet number varies with cell size (floc size in our case) and with different magnitudes of gravity. The grey region is where both diffusion and convection are important. Below the grey region, diffusion dominates, and above convection dominates. Increasing either gravity strength or cell size will cause the peclet number to increase, as is expected.

Figure 4.png


At this point it must be noted that all simulations thus far have been using a value of 9.8 m/s for gravity. If we re-ran the simulations using micro-g, 4 to 6 magnitudes weaker, all the effects of buoyancy would essentially be reduced to zero. This means that all cell and floc sizes would diffusion limited with peclet numbers approaching zero.

Conclusion

It has been observed that microorganisms behave and grow differently in a microgravity environment. These microbial alterations have implications that reach beyond crew health and into pharmaceutical and research domains. The leading theory for this change is the absence of buoyant forces limit mass transportation in space to diffusion alone. It was, therefore, pertinent to model both the diffusive and convective dynamics around a single cell. One of the main conclusions of Benoit et al 2008 was that for a single non-motile, 1 micron, E Coli cell on earth, the Peclet Number was such that the mass transport was dictated by diffusion alone. The paper goes on to say that a characteristic length of 100 microns was needed for the convective terms to play a significant role. The goal of this project was to first to recreate the Benoit et al model and produce similar results. This was accomplished using Matlabs PDEPE solver. The project obtained results on the same order of magnitude and deemed this a success. The project then went on to model the effects on the Peclet number when including E coli flocculation. The glucose consumption per cell was held constant and the flocs were assumed spherical. Models using both a minimum and maximum active floc biomass were used. It was found that flocs with a diameter above 20 microns had Peclet numbers large enough to indicate both convection and diffusion played an active role regarding on mass transport. This paper has helped produce evidence that while a single non-motile, 1 micron, E Coli cell on earth is dictated by diffusion alone, small bacteria flocs are effected by convective forces resulting from cell metabolism. When small flocs are exposed to microgravity these connective forces essentially disappear and the mass transport for the floc is dominated by diffusion alone, leading to the observed phenomena.

Now that biochemical convection has been shown to be a possible complicating factor to experimental design, further experiments can be designed. The difficulty of measuring the values discussed in this paper is the scale on which they vary. Boundary layer phenomena can have drastic bifurcations on length scales of microns, making accurate measurement often impossible. Hopefully as technology continues to develop, new techniques in micro and nano biology will be discovered.

References

  1. D.M. Klaus, M.R. Benoit, E.S. Nelson, and T.G. Hammond. Extracellular Mass Transport Considerations for Space Flight Research Concerning Suspended and Adherent in Vitro Cell Cultures. Bioserve Space Technologies, University of Colorado Boulder. 2004
  2. M.R. Benoit, R.B. Brown, P. Todd, E.S. Nelson and D.M. Klauss. Buoyant Plumes from solute Gradients generated by non-motile Escherichia Coli. Physical Biology 5 046007. 2008
  3. M.R.Benoit and D.M. Klauss. Microgravity, Bacteria, and the Influence of Motility. Advances in Space Research 39. 2007
  4. J. Bridgeman, B. Jefferson, and S.A. Parsons. Computational Fluid Dynamics Modeling of Flocculation in Water Treatment: A Review. Engineering Applications of Computational Fluid Dynamics, Vol. 3, No. 2, pp. 220-241.2009
  5. Tang, S., Y. Ma, and I. M. Sebastine. The fractal nature of Escherichia coli biological flocs Colloids and Surfaces B: Biointerfaces 20.3: 211-218.2001

Appendix

MatLab Code



%Robert Griffin Hale % Mathematical Biology


clc;clear all; close all tic


%Phisical constants and derived constants

   g=980; %[g/s^2]
   rho=1.0077; %[g/cm^3[
   visc=1.02e-2; %[g /cm ]
   k_visc= .01;  % [cm^2/s]
   Dg=6.8e-6; % Diffustion of Glucose
   Dw=1.21e-5;% Diffustion of waste/ acetate
   L=1e-4; %[cm] Diameter of single cell  this is 1 microns in diameter
   
   SA=(L/2)^2*pi*4;
   G_uptake=2.7e-16;% [g/s]'
   
   jg=G_uptake/SA; %[g cm^-2s^-1] 
   
   G0= 5e-3;  %  Bulk Glucose concentration 
   Bg=.34; % glucose expansion cofficent
   Bw=.04;% wasre expansion cofficent 

% PDEPE condtions m=2; % dictates sherical corinates


ri=L/2; % inner radius of a 10 micron bacteria cell

ro=1000*2*ri; % edge of boundry layer this is a "guess"

mpoint=100; % a hundred points

xmesh=logspace(ri,ro,mpoint);


tspan=linspace(0,60^2,60);  % time span


sol=pdepe(m, @pde1,@ic1,@bc1,xmesh,tspan);  % PDEPE


glucose=sol(:,:,1);  % first u is Glucose waste=sol(:,:,2);  % second is by product asummed acetate


%plot concentration glucose time and radial distance figure(1) surf(log(xmesh),tspan/60,(glucose))

xlabel('Log Distance [cm]') ylabel('time [min]') zlabel(' Log Concentration Glucose [g/cm^3]')

%plot concentration waste time and radial distance figure(2) surf(log(xmesh), tspan, log(waste))

xlabel('log distance') ylabel('time [s]') zlabel(' log concentration acetate ')


% Density changes

%change in density plot

delta_rho=(G0.*ones(length(tspan),length(xmesh))-glucose).*Bg+(waste*Bw);  % caclulates the change in densities of the


figure(3) hold on surf(log(xmesh), tspan/60,log(glucose)) surf(log(xmesh), tspan/60,log(waste)) xlabel('Log Distance [cm]') ylabel('Time [min]') zlabel('Log (u) Concentration [g/cm^3]')

legend('Glucose', 'Acetate')

figure (4)

surf(log(xmesh), tspan,(log(delta_rho))) xlabel('log distance') ylabel('time [s]') zlabel('change in density')


% caclulation of non dimetional coffiencts

g=980; %[g/s^2] gravity rho=1.0077; %[g/cm^3 average density of fluid visc=1.02e-2; %[g /cm ] k_visc= .01;  % [cm^2/s]



Gr=g.*L.^3.*(delta_rho/rho)/k_visc^2; % Grashof number Sc=k_visc/Dw;  %Schmidt number

V_max=sqrt(Gr./Sc).*k_visc./L;  % Max speed

Pe=V_max*L/Dg; %  Pe ?clet number 


figure(5)

surf(log(xmesh), tspan,((V_max))) xlabel('log distance') ylabel('time [s]') zlabel(' Max speed [cm/s]')


figure(6)

surf(log(xmesh), tspan/60,((Pe))) xlabel('Log Distance [cm]') ylabel('Time [min]') zlabel(' Pe')

%% Compare to paper

% compare max speed and max delta_rho to paper Maximum_rho=(max(max(delta_rho))) Maximum_speed=(max(max(V_max)))/100% (m/s) Max_PE=max(max(Pe))

toc


function [ pl,ql,pr,qr ] = bc1( xl,ul,xr,ur,t ) %UNTITLED4 Summary of this function goes here % Detailed explanation goes here

jg=-8.594e-9; %[g cm^-2s^-1] Glucose flux measured per cell with scale x 100 ja=jg*-.5; %[g cm^-2s^-1] Glucose flux measured from paper change


% LEFT

   %glucose
   pl(1,1)=jg;
   ql(1,1)=1;
   %waste
   ql(2,1)=1;
   pl(2,1)=ja;

% Right

   %Glucose


% no flux 
%glucose

pr(1,1)=ur(1)-(5E-3); % boundry condtion

qr(1,1)=0;


% no flux

% wasr

     pr(2,1)= 0; % boundry condtion 
     qr(2,1)=1;

end


function  [u0]  = ic1(x)

%Intial condtions

u0(1,1)= 5E-3;  % Bulk Glucose concentration

u0(2,1)= 0;  % intial acetate concentration

end


function  [u0]  = ic1(x)

%Intial condtions

u0(1,1)= 5E-3;  % Bulk Glucose concentration

u0(2,1)= 0;  % intial acetate concentration

end




Floc Pe clc;clear all; close all tic for j=1:10

%Phisical constants and derived constants

   g=980; %[g/s^2]
   rho=1.0077; %[g/cm^3[
   visc=1.02e-2; %[g /cm ]
   k_visc= .01;  % [cm^2/s]
   Dg=6.8e-6; % Diffustion of Glucose
   Dw=1.21e-5;% Diffustion of waste/ acetate
   L=j*10e-4; %[cm] Diameter of floc  this is 100 microns in diameter
   l=1e-4; % [cm] one cell with a diameter of 1 micron
   
   Rl=l/2;  % Radius of little cell
   
   RF=(L/2); %Radius Floc
   
   SA_F=RF^2*pi*4;
   
   G_uptake=2.7e-16;% [g/s]'
   
   
   % case 1
   V_floc=(4/3)*pi*RF^3;
   
   V_cell=(4/3)*pi*Rl^3;
   
   n1=V_floc/V_cell  % all floc is live cells 
   
   Flux_1= (n1*G_uptake)/SA_F %
   
   %case 2
   Ri=(RF-l)
  VI=(4/3)*pi*Ri^3;  % volume of dead cells 
  
  VM=V_floc-VI;  % colume of the living mantle 
  
  n2=VM/V_cell % living layer is one cell thick
  
  
  Flux_2= (n2*G_uptake)/SA_F


   G0= 5e-3;  %  Bulk Glucose concentration 
   Bg=.34; % glucose expansion cofficent
   Bw=.04;% wasre expansion cofficent 

% PDEPE condtions m=2; % dictates sherical corinates


ri=L/2; % inner radius of a 10 micron bacteria cell

ro=1000*2*RF; % edge of boundry layer this is a "guess"

mpoint=100; % a hundred points

xmesh=logspace(ri,ro,mpoint);


tspan=linspace(0,60*60,60);  % time span



sol=pdepe(m,@pde3,@ic3,@bc3,xmesh,tspan,[], Flux_1);  % PDEPE

sol2=pdepe(m,@pde3,@ic3,@bc3,xmesh,tspan,[], Flux_2);  %Optimize code later


% extract information glucose=sol(:,:,1);  % first u is Glucose waste=sol(:,:,2);  % second is by product asummed acetate


glucose2=sol2(:,:,1);  % first u is Glucose waste2=sol2(:,:,2);  % second is by product asummed acetate

% Density changes

%change in density plot delta_rho=(G0.*ones(length(tspan),length(xmesh))-glucose).*Bg+(waste*Bw);  % caclulates the change in densities of the

delta_rho2=(G0.*ones(length(tspan),length(xmesh))-glucose2).*Bg+(waste2*Bw);


% caclulation of non dimetional coffiencts

g=980; %[g/s^2] gravity rho=1.0077; %[g/cm^3 average density of fluid visc=1.02e-2; %[g /cm ] k_visc= .01;  % [cm^2/s]


Gr=g.*L.^3.*(delta_rho/rho)/k_visc^2; % Grashof number Sc=k_visc/Dg;  %Schmidt number

V_max=sqrt(Gr./Sc).*k_visc./L;  % Max speed

Pe=V_max*L/Dg; %  Pe ?clet number 

%  flux two optimize later 
Gr2=g.*L.^3.*(delta_rho2/rho)/k_visc^2; % Grashof number 

V_max2=sqrt(Gr2./Sc).*k_visc./L;  % Max speed

Pe2=V_max2*L/Dg; %  Pe ?clet number 


%% Compare to paper


% compare max speed and max delta_rho to paper Maximum_rho=(max(max(delta_rho))) Maximum_speed=(max(max(V_max)))/100% (m/s) Max_PE(j)=max(max(Pe)); Max_PE_2(j)=max(max(Pe2)); end %% close all; clc toc figure plot(((1:j).*10e-4)/1e-4,Max_PE,':b +') hold on plot(((1:j).*10e-4)/1e-4,Max_PE_2, ':b d')

plot(ones(1,11)*20,(0:10),'r') plot(ones(1,11)*60,(0:10),'r')

x=[20:length(Max_PE):60]'; %jbfill(x,[Max_PE(2:6)]',[Max_PE_2(2:6)]','b','b',0,1) %,edge,add,transparency)

xlabel('Floc Diameter [um]') ylabel('Peclet Number') legend('Fully Active Floc','Minimally Active Floc', 'Typical E Coli Floc size')
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