Gas Vesicle Structure: A physical analysis

Gas vesicles are proteineceous nano-structures with sizes in the sub-micron range making them a important exploitable tool in nano-sciences. A typical H. Salinarum gas vesicle measures 300nm in length and around 200nm laterally. A. flos-aquae vesicles are slightly larger in size but the culturing conditions required for the algae make them harder to extract. All the gas vesicles used in our flocculation experiments were extracted from H. Salinarum and were stripped of GvpC by using 6M urea lysis method. It becomes necessary for us to show that such a particle at room temperature is not a very potent floater and the steady state distribution is not good enough to allow considerable separation between the solution and the protein phase.(See "Analytical solution at steady state")

Electron micrograph of gas vesicles isolated from A. flos-aquae (left) and H. Salinarum (right). Taken from [1].

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 analysed by Strunk et al^[2]. As was noted, two of the lysines in the smaller alpha helix are solvent accesible. NHS-Biotin is a biotinylating reagent that reacts with primary amines (eg. lysines) in a peptide chain attaching a small spacer separated biotin moiety to them. The presence of a tetravalent biotin binding molecule (avidin/streptavidin) made us propose a biotin-streptavidin mediated strategy for increasing the effective hydrodynamic radius of these vesicles. It is interesting to note that the hydrophobicity of the internally exposed part of GvpA is what keeps the gases from diffusing out and leads to the formation of the gas filled cavity.

3D folded structure of gas vesicle forming protein A (Adapted from [2])

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^[3]) 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

Strunk and colleagues^[2] also proposed a plausible stacking pattern for the GvpA monomers that leads to the formation of the whole vesicle. According to their analysis, the best stacking pattern was obtained by the mutual docking of two molecules of the monomer followed by the stacking of these in three dimensions to form the vesicle (See figure).

Effective buoyant density

Most of the physical properties of the gas vesicles were uncovered by Walsby et al. in their 1994 review^[4] . 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.

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

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.

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.

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,

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

Where v_t is the terminal velocity and R_H 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. It is interesting to note that while the hydrodynamic radius is a linear function of the size of the gas vesicle, the volume scales as a cube of the radius. This leads to the logical deduction that the terminal velocity will increase if the particles are allowed to aggregate.

For a normal H. Salinarium gas vesicle, the terminal velocity comes out to be of the order of 10 nm/s . This is incredibly slow considering the size of a gas vesicle is an order of magnitude larger than this number.

Péclet number

The Péclet 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,

\[ P_e=\frac{Lu}{D} \tag{2.1} \label{eq:2.1} \]

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,

\[ D=\frac{k_{b}T}{6\pi \eta R_{H}} \tag{2.2} \label{eq:2.2} \]

Now, the terminal flow velocity from the last section can be used to calculate the Péclet number for this system. If the Péclet 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 at room temperature (T = 273K),
L ~ 500nm
u ~ 10nm/s
R_H ~ 200nm

Substituting these numbers in equation 2.2, we get a péclet number of the order of 10^-3, which implies dominant diffusive effects when gas vesicles are allowed to settle at the equilibrium concentration profile. In such a case, a model is needed that can account for the behaviour of floating particles in the highly diffusive regime.

Evolution of columnar concentration profile

The first step to solving any problem relating to diffusive transport is writing a convection-diffusion equation that takes all intricacies of the underlying system into account. For a system of sedimenting particles, it is given by^[5],

\[ \frac{\partial C}{\partial t}=\frac{\partial}{\partial h}(\frac{D \partial C}{\partial h})-\frac{\partial (C U(C))}{\partial h} \tag{3.1} \label{eq:3.1} \]

Where U(C) is the settling velocity of a particle in a region where the concentration is C. The fact that this velocity is related to diffusion arises due to long range (compared to particle size) velocity field interactions in the column. Two particles settling together tend to sediment faster compared to a single one due to the presence of fluid flow around the second one due to the motion of the first particle. If the suspension is sufficiently dilute and the particle sizes sufficiently small, these velocity fields can be ignored and each particle can be assumed to settle with a constant velocity (the terminal velocity). U(C) now reduces to -V_s, the terminal settling velocity (which is negative in this case because the particles are floating upwards).

The translational diffusion coefficient is not a height dependent parameter and can be taken out of the partial differential term giving the following equation in absence of an underlying velocity field,

\[ \frac{\partial C}{\partial t}=D \frac{\partial ^{2} C}{\partial h^{2}}+V_{s} \frac{\partial C}{\partial h} \tag{3.2} \label{eq:3.2} \]

This equation can be solved either analytically or numerically to yield solutions as follows.

Analytical steady state solution

The steady state solution can easily be calculated by setting \(\frac{\partial C}{\partial t}\) to zero. This solution of the resulting equation is of the form,

\[ C(h)=C_{1} e^{kh}+C' \]

Putting this in the steady state equation yields \(k=\frac{-V_s}{D}\). The value of \(C_1\) and \(C'\) can be found by normalising this answer with the total amount of gas vesicles in the column and applying the necessary boundary conditions .

Illustration 1: Steady state solution for a column

We assume the boundary conditions to be such that \(C'=0\), \(C_1\) can be calculated by normalising the function over gas vesicles in the column.

\[ \int_{0}^{L} C_1 e^{\frac{-V_s h}{D}} A dh = N \]

Where A is the area of the column and N the total number of gas vesicles in the vessel. The normalised equation is,

\[ C(h)=\frac{V_s N}{AD(1-e^{\frac{-V_sL}{D}})} e^{\frac{-V_s h}{D}} \]

We take the following parameters to simulate a 1 cm² cuvette with solution filled up to a height of 2 cm ,

\[V_s \sim \frac{4gR_H^2}{18\eta} (\rho-\rho_{gv})\] \[A \sim 1 cm^2\] \[D = 6\pi \eta R_H \] \[L \sim 2cm\]

To take into account the dependence of terminal velocity on hydrodynamic size, we have considered the gas vesicle to be a sphere of a radius equal to the hydrodynamic radius measured in DLS. The steady state profiles for some effective \(R_H\) are given below,

Figure 1: Gas vesicle concentration profile at steady state (C₀ is the mean concentration in the column)

These curves make it quite clear that unaided gas vesicles (\(R_H \sim 100nm\)) have an almost uniform distribution across the length even at the steady state which is achieved after an infinitely long span of time. This was the first insight that made it clear that to make any efficient use of a gas vesicle's buoyant properties, a strategy was required that would increase the \(R_H\) considerably and would allow them to float better.

Numerical Solutions

A numerical solution requires intelligent initial and boundary condition choices to give reasonable results. Computational tools^[6] were used to solve the above equation with different nature conditions to see how the distribution changes over time. Some of the plausible boundary conditions and their qualitative solutions are given in the illustrations section of our model.

Note that these examples are only qualitative and only give a flavor of how the concentration profiles evolve over time. All the parameters in the primary differential equation have been set to unity to get the plots. We wish to show how these plots can be used to find the functional forms of the OD vs time curves that are obtained in the spectrophotometry assay for gas vesicles.

Illustrations: Qualitative solutions to the model

Illustration 2: Uniform initial profile with exponential boundary conditions.

Assuming the initial mixing is uniform, this is the most natural way in which we expect the concentration function to evolve on the boundaries. The exact initial and boundary conditions are taken such that they satisfy each other nicely. The contour plot below shows how the profile evolves over time. The legend on the right shows the relative magnitude of concentrations present in the plot.

Figure: Uniform intial profile with exponential saturation at boundaries

Illustration 3: Exponential initial profile with exponential boundary conditions.

Such a situation can arise in the event the particles have settled down due to gravity initially but start floating instantaneously at t=0. While quite unrealistic, this solution might be useful in cases where the gas vesicles have been reversibly denatured but are allowed to float up after removing the denaturing agent. The plot below shows how the profile evolves over time.

Figure: Exponential intial profile with exponential saturation at boundaries

Illustration 4: Linear initial profile with exponential boundary conditions.

This is the most unlikely of the three cases we've dealth with till now. An linear initial profile is almost impossible under normal conditions unless some kind of external agency maintains it. Even then, we've tried to solve the equations to show how the concentrations will evolve for such a case.

Figure: Linear intial profile with exponential saturation at boundaries

Illustration 5: Uniform initial profile with quadratic boundary conditions.

While not usually encountered in nature, saturating inverse quadratic solutions are given here for demonstration purposes. The first of them is obivously, the uniform initial distribution.

Figure: Uniform intial profile with quadratic saturation at boundaries

Illustration 6: Exponential initial profile with quadratic boundary conditions.

Figure: Exponential intial profile with quadratic saturation at boundaries

Illustration 7: Linear initial profile with quadratic boundary conditions.

Figure: Linear intial profile with quadratic saturation at boundaries

References

[1] Daviso E, Belenky M, Griffin RG, Herzfeld J. Gas vesicles across kingdoms: a comparative solid state NMR study. Journal of molecular microbiology and biotechnology. 2013;23(0):10.1159/000351340. doi:10.1159/000351340.
[2] Strunk, T., Hamacher, K., Hoffgaard, F., Engelhardt, H., Zillig, M. D., Faist, K., Wenzel, W. and Pfeifer, F. (2011), Structural model of the gas vesicle protein GvpA and analysis of GvpA mutants in vivo. Molecular Microbiology, 81: 56–68. doi:10.1111/j.1365-2958.2011.07669.x
[3] Buchholz B, Hayes P, Walsby A. Microbiology 139(10):2353-2363 doi:10.1099/00221287-139-10-2353
[4] Walsby AE. Gas vesicles. Microbiological Reviews. 1994;58(1):94-144
[5] Nielsen, Peter. Combined Convection-diffusion modelling of sediment entrainment. Coastal Engineering 1992 . doi:10.1061/9780872629332.244
[6] Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL (2012).