1. Gas Vesicle Structure
2. Mathematical Model
3. Experimental Data
4. Results
5. References

Gas Vesicle Structure

Gas vesicles are proteineceous nano-structures that are utilized by many aquatic micro-organisms like halo-bacteria and some algae to provide buoyancy. The structure and arrangement is highly conserved between organisms with width being almost the only widely varying parameter. They contain gases which diffuse in during formation and are kept localised by the hydrophobicity of the inner membrane. Unlike true vesicles, these are made of proteins instead of phospholipids and are hence of considerable interest. Each gas vesicle is composed of two primary protein monomers, the gas vesicle forming proteins A (GvpA) and C (GvpC). The entire structure will be discussed in the following sections.

Verification of presence of Gas Vesicles

The easiest way to assay presence of gas vesicles is their disappearance under high pressure under a microscope. This was observed even during normal experiments. Fully filled micro-centrifuge tubes containing dilute gas vesicle suspensions lost their faint opalescence when the tube was closed (this did lead to a loss of samples). A more strict assay was done using DLS (See Dynamic Light Scattering) and SEM Imaging to pinpoint the exact size of the nano-particles. It was found that these gas vesicles have an effective hydrodynamic radius of around 230nm. This estimate was particularly valuable in the development of our model.

GvpA

The gas vesicle forming protein A (~8kDa) forms the main ribbed structure of the gas vesicle. The 3-Dimensional structural model of the protein was recently decoded by Strunk et.al ([1]). As was noted, the best probable docking pattern in a vesicle involves formation of a dimer and their consequent stacking. We intended to re-create the whole gas vesicle structure from this data and try to explain the growth pattern noticed during their formation. It is interesting to note that the hydrophobicity of the internally exposed part of GvpA is what keeps the gases from diffusing out.

GvpC

The gas vesicle forming protein C is the second most abundant protein found in the nanostructure. It is still far rarer compared to GvpA (1:25 molar ratio, [2]) and does not contribute to the structure but rather to the integrity of the gas vesicle. The vesicles used in the various experiments were stripped of GvpC before use and hence were much more sensitive compared to wild type gas vesicles. The critical pressure for the gas vesicles was found to reduce drastically on the removal of this protein.

Stacking of proteins in a GV nanoparticle

3-Dimensional structure of the nanoparticle

Effective buoyant density

Most of the physical properties of the gas vesicles were uncovered by Walsby et al. In their groundbreaking review ([3]). They calculated the average buoyant density of a gas vesicle to be around 120kgm-3. Note that while this density if much less than that of water, it doesn’t necessarily imply a flotation advantage due to the presence of diffusive forces in a suspension. The smaller the particle, higher is the magnitude of these diffusive interactions and more is the deviation from the ideal concentration profile in a column. The mathematical model part of our project deals with finding how these profiles evolve for overtime and how this evolution changes when we purposefully make them flocculate using agents like chitosan and biotin-streptavidin.

Mathematical Model

The following sections detail the development of our model which deals with the dynamics of gas vesicle motion in a medium and how their concentration profiles change over time.

Terminal Velocity

A floating particle in a media experiences multiple types of forces, some of which are more prominent than others. Given the buoyant density of the particle, we can find the terminal velocity by equating all the upward forces to the downward ones and solving for velocity from the stokes’ drag term.

[SCHEMATIC OF A GAS VESICLE]

Assuming the gas vesicle to be of the shape given above, its volume in terms of length l, half angle θ and radius r is given by,

\[ V = \pi r^{2}(l+\frac{2}{3}r cot \theta) \tag{1.1} \label{eq:1.1} \]

So, the buoyant force on the gas vesicle is given by,

\[ F_b=V\rho g = \rho g\pi r^{2}(l+\frac{2}{3}r cot \theta) \tag{1.2} \label{eq:1.2} \]

Where ρ is the density of the media.

Similarly, the force due to gravity is,

\[ F_w=V\rho_{gv} g = \rho_{gv} g\pi r^{2}(l+\frac{2}{3}r cot \theta) \tag{1.3} \label{eq:1.3} \]

Where ρ_gv is the buoyant density of the gas vesicle.

When the particle reaches its terminal velocity, all the forces are balanced. Thus we can equate,

\[ F_{d} + F_{w} = F_{b} \tag{1.4} \label{eq:1.4} \]

The viscous drag is usually a complex function of particle shape and size, in our case however, the DLS experiment directly gives the hydrodynamic radius of the particle. Hence we can use the well known expression for drag on a spherical particle given by,

\[ F_{d}=6\pi\eta r v \tag{1.5} \label{eq:1.5} \]

Where v_t is the terminal velocity and r the hydrodynamic radius.

Solving for v_t, we get

\[ v_{t}=\frac{Vg}{6\pi\eta R_H} (\rho-\rho_{gv}) \tag{1.6} \label{eq:1.6} \]

The various parameters can be modified to find the terminal velocity for a particular kind of nanoparticle after measuring the hydrodynamic size with a DLS system.

For a normal H. Salinarium gas vesicle, the terminal velocity comes out to be,

[CALCULATE]

Peclet number

The Peclet number is a dimensionless quantity that is used to determine the relative magnitudes of advective and diffusive transport phenomena. In the case of a particle undergoing flotation(or sedimentation), it can be easily calculated by taking the following ratio,

[EQUATION 2.1]

Where L is the characteristic length scale in the system (~200nm in the case of a gas vesicle), u the local velocity of the fluid and D the mass diffusion coefficient.

The Einstein relation for a spherical particle gives the Diffusion coefficient from the viscosity of the medium through the following relation,

[EQUATION 2.2]

Now, the terminal flow velocity from the last section can be used to calculate the Peclet number for this system. If the Peclet number is extremely small compared to 1, diffusion is dominant over advective transfer and must be considered during our calculations.

For a typical gas vesicle particle,

Evolution of gas vesicle concentration profile in a column

Experimental Data

Visual Analysis

Flotation Spectrophotometry

Chitosan

Double replicates of three different concentrations of chitosan were used with gas vesicles (30ul stock) and the resulting solutions were diluted to 2ml to perform a flotation spectrophotometry assay.

Tube Label	Effective gas vesicle concentration	Effective chitosan concentration	Remarks
1	15 ng/μl	0 ng/μl	Control tube
2A	15 ng/μl	5 ng/μl	First replicate
2B	15 ng/μl	5 ng/μl	Second replicate
3A	15 ng/μl	50 ng/μl	First replicate
3B	15 ng/μl	50 ng/μl	Second replicate
4A	15 ng/μl	500 ng/μl	First replicate
4B	15 ng/μl	500 ng/μl	Second replicate

The data from the spectrophotometer assays for chitosan can be found here.

An analysis of the data is given in the results section

Electron microscopy

Dynamic Light Scattering

Results

Chitosan

The data when averaged over the replicates shows a marked faster decrease in OD as time passes for chitosan treated gas vesicles. The gas vesicles without chitosan show no significant decrease over a duration of two hours while the ones with the maximum chitosan concentration show a fast decrease at the start which saturates as time passes. All other curves lie in an intermediate region. (Fig 1) The plot was smoothed out over a window of 85 data points giving the smooth profiles. (Fig 2) [[File:T--IISc-Bangalore--Model-resultsChitosanFig1.png]] [[File:T--IISc-Bangalore--Model-resultsChitosanFig2.png]]