Virtual Sponge Models

Introduction

In this section we model the water flux in living sponges. As we could not test our constructs in living sponges (sponges are fragile organisms, bringing them to paris and then growing them is complicated), we relied on models in order to find out: what would be the effectiveness of our constructs in living sponges ?

More precisely we are interested in finding: what is the quantity of compound that is in contact with our bacterium ?

With this quantity and the number of bacteria in the sponge, we can the connect with the models developed for PCB and phenol (TODO) sensing and then relate the sensing capacity to the concentration of compound in the surrounding water.

To answer this question, we built two different model :

1. A simple model where we consider only water flows without trying to take into acount the very specific geometry of the sponge.
2. A 2D diffusion model where we take into acount the geometry of the sponge.

Throughout this section we base our study on the Spongia Officinalis species, because i) there are evidences that pseudovibrio bacteria live inside (TODO: cite) and ii) its is a quite common type of sponge.

Model 1: Simple Fluxes

For this model, we make simple computations based on the intake and expeled quantities of water. Our main assumptions are the following :

1. Bacteria are uniformely distributed in the sponge
2. Compounds diffuse instantly inside the sponge (the quantity of compound in the same everywhere inside)

These two assumptions imply that each bacterium is in contact with the same quantity of compound.

Model Formulation

We call φ_in (in ml/h/cm³) the quantity of compounds (<= 0.2 μm) filtered by a sponge of volume V (cm³). The compound is present at concentration C (mol.ml^-1). Then the quantity of compound in contact with the bacteria, Q (mol.h^-1), is:

Q = φ_inVC

Parameters

Name	Value	Unit	Ref
φin	[0.1-0.3]	cm3(water).cm-3(sponge).s-1	WATER TRANSPORT, RESPIRATION AND ENERGETICS OF THREE TROPICAL MARINE SPONGES
V	[66.8-116]	cm3	Filtering activity of Spongia officinalis var. adriatica (Schmidt) (Porifera, Demospongiae) on bacterioplankton: Implications for bioremediation of polluted seawater

Results

We present in Figure1 the results obtained for different parameter sets (min, mean and max values). The value of Q increases linearly with the concentration and is in the range of the μmol.

The equations for the three lines in Figure1 are the following:

• Red: y = 0.00668x
• Green: y = 0.01828x
• Blue: y = 0.0348x

The code was realized in Matlab and is available here.

Model 2: 2D diffusion

For this second model we want to take into account the geometry of the sponge. For the equations and geometry to be tractable we will consider a 2D slice of a sponge. Our assumptions are the following :

1. Bacteria are uniformely distributed in the sponge (as for model 1)
2. Sponge geometry is approximated as a sphere
3. The interior of the sponge is a uniform medium in which the compound diffuse isotropically with a coefficient D

We emphasize that assumption 2 may not be a pure mathematician's idealization, some spongia officinalis sponges have approximately spherical shapes as presented in Figure2a (Although others do not, a wide variety of shapes depending on the environment are exhibited). Some articles also represent some species of sponges as ellipsoidal, see Figure2 b (TODO cite).

Geometry

For the simplicity of the simulations, we represent only a slice of a spherical sponge, the slice going through its center. We thus have the following geometry presented in Figure3. Orange regions represent the osculum where output water flow occurs and green regions represent regions containing ostia allowing intake of water. The sponge is fixed to the ground in the black region, preventing water flow in this region (reflecting boundary). It is possible to add as many oscula as wanted and to tune their size.

Model Formulation

We model the diffusion of the compound in the sponge by the following classical diffusion equation:

Where:

• φ(r,t) (mol.) is the quantity of compound per hour at point r and time t
• D (cm.s^-1 is the coefficient of diffusion
• Δ² is the 2D laplacian

In the simulations we are interested in the steady state distribution of compounds which is lim_t->∞ φ(r,t). Note that as we have output flows in the model, the distribution at steady state will not be uniform.

We model the different parts of the sponge's surface by different boundary conditions :

• Ostia: positive Dirichlet boundary condition
• Oscula: negative Dirichlet boundary condition
• Ground: Neumann boundary condition ∂φ(r,t)/∂n = 0 (n is the normal vector).

Parameters

The model parameters are the following:

• R (cm): radius of the circle
• D (cm².s^-1) diffusion coefficient of the compound
• φ_in (cm³) expellent water flow (φ(r,t) = φ_in on ostia)
• φ_out (cm³) inhalent water flow (φ(r,t) = φ_out on oscula)
• N_osc: number of oscula
• W_osc: width of the oscula

It is also possible to tune the width of the ground area but we will let this parameter constant throught our simulations.

Here are the formula we used to determine the model parameters

• R, is obtained using the formula for the volume of a sphere: R = (0.75 * V/π)^1/3
• φ_in is the quantity of water entering the sponge in 1 hour.
• φ_out is the quantity of water leaving the sponge in 1 hour.

We take the following parameter values:

Name	Value	Unit	Ref
φin	[6.68-34.8]	cm3(water)	computed from previous φin and V
φout	-φin	cm
R	[2.517-3.025]	cm	computed from V
Wosc	[0.3-1]	cm	FAO (http://www.fao.org/docrep/field/003/AC286E/AC286E01.htm)
D	5	cm2.s-1

Results

Here are some preliminary results using the minimal parameter values: This simulation was conducted using freefem++, the code is available here.

Conclusion

It is important to take into account the geometry of the sponge and the number and size of oscula.