
MBW:Governing Fluid Dynamics of NonMotile Bacteria in Varying Gravitational DomainsFrom MathBioContentsExecutive SummaryIt is difficult in space to conduct a truly controlled biological experiment. Many properties of space flight, specifically microgravity, have yet unknown effects on organisms. Variations can be caused to the interior of a cell, the microenvironment directly surrounding the cell, and the macroscale containingmedium. 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 microg on the chemical concentrations in boundary fluid layer surrounding a cell will be discussed. IntroductionHistoryBeginning 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 causeandeffect pathways have been hypothesized and researched including increased radiation levels, hyperg during launch, randomg (vibration) and obviously microg. Microgravity is defined as gravity with magnitude that is more than two levels of magnitude below 1g baseline. Typically gravity in orbit has a magnitude of E4 g ^{[2]}. This paper will concentrate explicitly on single celled organisms, such a simple bacteria or protozoa; and how their behavior is altered by microg 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 microg, 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 reentry. 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 byproducts will be produced in space than on earth, and the cells will grow larger. This seems counterintuitive, because as we will later discuss, masstransport (and therefore the supply of fuel) around cells decreases in microg. However, the phenomena can be explained similarly to the effect of batchfeeding 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 overfed, they have no stimulus to work efficiently. Most effects of microg are characterized in comparison to the 1g equivalent system. They can either be observed in a true microg environment, or some simulation thereof (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 EffectsDirect 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 (microg, 1g, hyperg, randomg) 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 1g and microg. 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 1g, without the hazard of rupturing. Indirect Gravitational EffectsIndirect effects of microg 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 microg. 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 DesignMuch of the difficulty in interpreting data taken in space comes from the variability added when adapting an experiment for a microg 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. EquipmentOften, scientists will try to simulate microg 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 infact 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 lessdense fluids ^{[3]}. ModelModel DevelopementThere 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 biproduct (note the subscript i's). It should be noted that this model only considers the consumption of glucose and assumes the biproduct is solely acetate; produced at a rate of half the glucose consumed by the cell.
where
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. Parameters, Measured Values, and Boundary ConditionsWe 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.
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. Adapting Model for FlocculationThus 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, . 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 can be calculated as by dividing the total consumption rate, by the total outer surfacearea of the floc. This assumes the floc is strictly spherical, and of radius .
This means we can approximate the maximal boundary glucose flux, , and the minimal flux, , if we can calculate the number of active cells, n, in each case.
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 DiscussionOriginal ModelIn 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, nonmotile 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, , , and agreed within roughly 25% error. The comparison of results from 1g simulations are shown below.
Next (below) is a similar plot showing both the glucose concentration as well as the acetate (byproduct) 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. Finally, we plotted the Peclet Number varying in space and time. Note the magnitudes of Pe is in the range of (e3), 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. Adding FlocculationKnowing 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. 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 masstransfer 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.
ConclusionIt 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 nonmotile, 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 nonmotile, 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
AppendixMatLab Code
%Phisical constants and derived constants g=980; %[g/s^2] rho=1.0077; %[g/cm^3[ visc=1.02e2; %[g /cm ] k_visc= .01; % [cm^2/s] Dg=6.8e6; % Diffustion of Glucose Dw=1.21e5;% Diffustion of waste/ acetate L=1e4; %[cm] Diameter of single cell this is 1 microns in diameter SA=(L/2)^2*pi*4; G_uptake=2.7e16;% [g/s]' jg=G_uptake/SA; %[g cm^2s^1] G0= 5e3; % Bulk Glucose concentration Bg=.34; % glucose expansion cofficent Bw=.04;% wasre expansion cofficent % PDEPE condtions m=2; % dictates sherical corinates
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
%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
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=1.02e2; %[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]')
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=8.594e9; %[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)(5E3); % 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)= 5E3; % Bulk Glucose concentration u0(2,1)= 0; % intial acetate concentration end
function [u0] = ic1(x) %Intial condtions u0(1,1)= 5E3; % Bulk Glucose concentration u0(2,1)= 0; % intial acetate concentration end
%Phisical constants and derived constants g=980; %[g/s^2] rho=1.0077; %[g/cm^3[ visc=1.02e2; %[g /cm ] k_visc= .01; % [cm^2/s] Dg=6.8e6; % Diffustion of Glucose Dw=1.21e5;% Diffustion of waste/ acetate L=j*10e4; %[cm] Diameter of floc this is 100 microns in diameter l=1e4; % [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=2.7e16;% [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=(RFl) VI=(4/3)*pi*Ri^3; % volume of dead cells VM=V_flocVI; % colume of the living mantle n2=VM/V_cell % living layer is one cell thick Flux_2= (n2*G_uptake)/SA_F
G0= 5e3; % Bulk Glucose concentration Bg=.34; % glucose expansion cofficent Bw=.04;% wasre expansion cofficent % PDEPE condtions m=2; % dictates sherical corinates
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
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
% 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=1.02e2; %[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
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')
