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Gas Vesicle Structure
Terminal Velocity
Péclet Number
Concentration Profiles
Illustrations
Optical density estimates
References

Gas Vesicle Structure: A physical analysis

Our Systems

In our lab, we have access to purified gas vesicles from Halobacterium salinarum NRC-1 and an agar slant of Anabaena flos-aquae. Our models will use typical values of physical parameters available in literature for gas vesicles from these species.

Gas vesicle proteins

GvpA

GvpA is a short (72 aa, 8 kDa) protein that polymerizes to form the main "ribbed" structure of the gas vesicle. The abundance of hydrophobic beta chains (45 residues) in GvpA makes the protein extremely averse to water; indeed, this hydrophobicity prevents water molecules from entering the interior of the gas vesicle during its formation, allowing it to fill with gas instead. Note that two lysine residues in the smaller α-helix are solvent-accessible.

Stacking of GvpA monomers

Strunk et al.^[2] 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).

GvpC

GvpC is the second-most abundant protein found in gas vesicles but it is still far rarer than GvpA (1:25 molar ratio^[3]) and does not contribute to the structure but rather to the integrity of the gas vesicle! Gas vesicles stripped of GvpC have a drastically-lower critical pressure than wild-type gas vesicles and are much more sensitive to mechanical collapse.

Effective buoyant density

The average buoyant density of a gas vesicle is reported to be around 120 kgm^-3. While this density is much lower than that of water, this doesn’t necessarily imply a flotation advantage to the gas vesicle due to the presence of diffusive forces in a suspension. The smaller the particle, the stronger these diffusive interactions and the greater the deviation from the ideal concentration profile in a vertical column.

Our Model

The following sections detail the development of our mathematical model, which deals with gas vesicle dynamics in a medium and determines how their concentration profiles in a suspension evolve over time.

Terminal Velocity

A floating particle in a medium experiences multiple forces, some more prominent than others. Given the buoyant density of the particle, we can calculate its terminal velocity by equating all the upward forces and the downward ones and solving for velocity from the Stokes’ drag term.

Modeling the gas vesicle as displayed above, its volume V in terms of length l, half-angle θ and radius r is given by

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

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, we will evaluate the drag using the hydrodynamic radius of the particle using the well-known expression for 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

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

For a typical haloarchaeal gas vesicle, the terminal velocity is on the order of 10 nm/s, which is incredibly slow considering that a gas vesicle moving at this speed will take about thirty seconds to cover its own length!

Note that this is a very bare-bones approach to calculating the terminal velocity of a gas vesicle which is only an order of magnitude factor in our other calculations. A more detailed analysis was done by the UAlberta 2017 iGEM regarding the flotation of cells containing gas vesicles. The latter part of their model was borrowed from our own analysis on gas vesicles in a diffusive regime.

Knowing the hydrodynamic radius of the particle, we can modify the various parameters to determine its terminal velocity. Notice an interesting fact: while the viscous drag scales with radius, the buoyant force scales with volume! Even this preliminary modelling is sufficient to decide the direction of our project: to make gas vesicles float better, we need to pack them together in a small volume — we have to aggregate them!

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, it can be 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 of the system (~200 nm for 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 ~ 500 nm
u ~ 10 nm/s
R_H ~ 200 nm

Substituting these numbers in Equation 2.2, we get a Péclet number on 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, we require a model that can account for the behavior of floating particles in the highly-diffusive regime.

Evolution of columnar concentration profile

The first step in solving any problem related to diffusive transport is to write 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 as a result of 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 our situation 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

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

Substituting this in the steady state equation yields \(k=\frac{-V_s}{D}\). The value of \(C_1\) and \(C'\) can be found by normalizing this answer with the total number 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 normalizing the function over the number of 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 normalized equation is

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

a

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,

These curves make it quite evident 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 yield reasonable results. Computational tools^[6] were used to solve the above equation with different conditions to see how the distribution changes over time. Some 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 obtain 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.

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.

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.

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.

Illustration 6: Exponential initial profile with quadratic boundary conditions.

Illustration 7: Linear initial profile with quadratic boundary conditions.

Optical density estimation

The main reason to build a model around flotation was to check how gas vesicle flotation is aided by our experiments. This itself was shown by the steady state solution which gave a realistic estimate of how the concentration profile would look after a long time. The time series solutions however, give an insight on how to optical density of the medium would change had the column been in a spectrophotometer taking readings at regular time intervals. Experimental data obtained from this experiment can be found in the "Experiments" section of this wiki. Here, we wish to show how a rough functional form of the optical density time series can be obtained from this model.

Note: We consider only the initial and boundary conditions from the first illustration for this estimation

Assuming the photodiode in the spectrophotometer reads only over this range, intensity in this region will be determined by the average concentration of the gas vesicles in the suspension. This is bascially an integral of the concentration over the sampling region volume determined by the volume itself

\[ C_{avg}(t)=\frac{\int_{l_0}^{l_1} C(h,t) A dh}{A(l-l_0)} \]

Numerical integration was done over different sampling region lengths (same right hand limits) to get the following curves showing how the average concentration changes with time

Assuming the Optical density is proportional to concentration, the percent change in OD can be obtained from this data to get a qualitative function that can be fit to our data from the spectrophotometric assay,

\[ \textrm{percent change in OD}=\frac{(C_{avg}(t)-C_{avg}(0)) 100}{C_{avg}(0)} \]

Note that this operation will only shift the curves down to zero and will scale them up. Using this formula, the following curves are obtained.

These are indeed similar to the ones we obtained in our spectrophotometeric experiments. These curves, though obtained numerically, can be used to fit to the data and obtain estimates of particle sizes and the diffusion coefficient.

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).

Team:IISc-Bangalore/Model