IGEM:IMPERIAL/2006/project/Oscillator/Design

The Lotka-Volterra Model and Its Consequences

Figure 1. A block diagram of Predator-Prey Interactions with A being the prey and B being the predator. Predator eating prey (modelled with A+B) leads to progeny of B, whilst A continuously reproduces exponentially.

Predator-prey interactions form stable biological oscillations with frequency which can be calculated with precise frequency given certain assumptions. We decided to model our biological oscillator on the basis of predator-prey interactions hoping that this novel approach will lead to success.

In order to ensure that the concentration of our molecules outside the cell will be fluctuating, we must have a multi-cellular system. The flaw in previous relaxed oscillator models was that it was based upon a single-cellular culture, unable to control the entire population effectively. By having a two cell design, then we force molecules to interact outside of the cell, thus changing the concentration of the extracellular medium.

The key assumptions to the Lotka-Volterra model are as follows:

• Predator and prey distributions are random.
• The only cause of death to the prey species is through predation
• The growth rate of the predator species is solely dependent on predation of the prey.
• The Prey species shows exponential growth in the absence of predation with no carrying capacity
• The Parameters are constant over time

Only once all of these assumptions are met can biological oscillations occur.

Since chemical species are more controllable than biological ecosystems, it should, in theory, work better in the laboratory. However, oscillatory chemical reactions will eventually die down due to the second law of thermodynamics, which states that the energy can neither be created nor destroyed. Oscillatory chemical systems pose a sort of Holy Grail for chemistry, since their existence would signify a perpetual motion system enabling energy to be created continuously without input. The closest chemical system which approximates oscillatory movement is the Belousov-Zhabotinsky (BZ) reaction. This reaction oscillates between cerium oxidations states III and IV, producing a remarkable change in colour. These oscillations can be sustained for approximately an hour before dying down with a frequency which is dependent upon the concentrations of the reactants in the system.

Biological systems offer a completely different perspective on oscillatory behaviour since the oscillatory movements will no longer be bound by the second law of thermodynamics. The BZ reaction will theoretically be continuous if we are able to sustain the reactants of the system. However, in biological systems, we are able to feed the organisms to continuously produce the oscillatory behaviour, so we can inherently maintain the oscillatory behaviour. If we are considering Escherichia coli (E. coli) as our species, then all that is required is to keep the media supply running with constant oxygenation. Now all that is required is to find molecules/proteins which will mimic the behaviour of species A and B in the Lotka-Volterra model.

Quorum Sensing/Quenching

It is desired to have a molecule that easily passes through the membrane in order for the system concentration to change. In comparison to the Lotka-Volterra model, then we need the plasma membrane of E. coli to be permeable to molecules A and B. Quorum sensing/quenching molecules offer good candidates for molecules A and B. Their small molecular structure enables it to easily pass through the cell and affect the transcription of another cell, so it necessarily has to exit the cell.

Figure 2. N-acylhomoserine lactone (AHL) molecule, typical quorum sensing molecule in gram negative bacteria

Our predator can then be the AHL degradation enzyme which will “eat” the prey (AHL), thus can no longer be active as a signalling molecule.

Figure 3. The degradation action of AHL-lactonase and AHL-acyclase, two different methods of quorum quenching

By using AHL as our prey and an AHL-lactonase or AHL-acyclase as our predator in the Lotka-Volterra model, then it is only required to fiddle with the expression of the two in the cell in order to produce oscillatory movements. We must remember that the key assumptions listed above (now applied to chemical species), must still hold for the model to operate.

Possible candidates for the predator in our biological model are quorum quenching enzymes such as AiiA and AiiD. In a paper published in 2005 by Yi-Hu Dong et al., they investigated several quorum quenching enzymes discovered throughout the years and compared their activities. aiiA, by far, was the most general of AHL-acylase enzymes, whose effect was seen in both gram negative and gram positive bacteria. As seen in Figure 3, acyclase enzymes hydrolyse the amine group in the AHL molecule, rendering the products useless in cell to cell communication.

System Design

The main system design consists of a black box with the output as a sinusoidal (oscillating) signal. The input will not be required, since we assume that there will be inherent noise in the system to set the system in motion. However, when modelling a perfect system, transcription must be activated first, but this is discussed further in the modelling document.

Figure 4. System design view, showing sinusoidal output

Within the system, we have our predator and prey cell which will be producing molecules A and B.

Figure 5. General design level diagram implementation of the Lotka-Volterra model within the system using two molecules, A and B

We will be monitoring the concentration of molecule A. Within our system, cell A produces molecule A which gets “eaten” by molecule B. The AB complex then activates cell B to produce more molecule B. We then came up with two different methods of making a positive feedback mechanism to produce molecule A to promote the exponential growth required by the Lotka-Volterra predator-prey model.

After looking at possible molecules of A and B, we have decided that molecule A will be AHL and molecule B will be AiiA, the quorum quenching enzyme.

Figure 6. Part diagram of cell A (prey producing cell) using a constitutive promoter and positive feedback

The constitutive promoter (Pc) continuously produces LuxR, a receptor for molecule A (AHL). Initial stimulation by AHL will result in the formation of an AHL/LuxR complex which will activate transcription at the promoter pLux. Consequently, AHL is produced from the cell. After the initial stimulus, production of AHL continues as it binds automatically to the pLux promoter, not needing another injection of AHL.

Figure 7. Part diagram of cell A (prey cell), stimulated by pLux promoter

In this design, the production of both LuxR and LuxI (AHL) will self-stimulate the production of more LuxI, contributing to the positive feedback loop. Using CellDesigner software, we are able to model each and determine the characteristic output which would result in a perfect system without noise. More information on this is contained within the modelling document.

After we are able to produce a exponential growth in the prey, we need to create a complementary predator design. Keeping in mind that the predator should combine with the prey and stimulate the production of more predator (assumption in Lotka-Volterra model), we are again able to use LuxR as an internal receptor and aiiA to degrade the AHL inside the cell. One must remember that the action of LuxR and aiiA are contained within the cell. If we are able to create a concentration gradient across cell B, then the diffusion across the membrane will be proportional to the concentration outside the cell, even with desired oscillations.

Figure 8. Cell B (predator cell) showing influx of molecule A and subsequent degradation

As seen in Figure 8 above, we can use aiiA as our predator to consume the population of molecule A (AHL). By degrading AHL, we increase the concentration gradient between the outside and inside of the cell, resulting in the decrease in the concentration of A outside the cell. Once the gradient has reached the low threshold, production of aiiA will stop, enabling the build-up of AHL outside the cell again. Once cell A and B are activated to work together, then this oscillatory concentration change of AHL theoretically would continue until nutrients are depleted. If we keep these cells in a bioreactor, removing dead cells and replenishing the medium, we can sustain these oscillations until we stop the bioreactor.

We should note that aiiA and LuxR are fully contained within the cell and are unable to diffuse through the membrane. AHL, on the other hand, is very permeable, and diffuses readily through the membrane.

This design also has the strength that there are already working parts from the Registry of Standard Biological Parts (Registry) that we can use.