Team:Waterloo/Math Book/Conjugation

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<td>Agent division frequency* (Unit: division / h)</td>
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<td>Agent division frequency* (Unit: divisions / h)</td>
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<td>0.0011 nM &bull; min<sup>-1</sup></td>
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<td>Donor: 0.5, Recipient: 0.5</td>
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<td>mRNA production from SarA P1 Promoter</td>
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<td><i>S. aureus</i>: 1.43, <i>E. coli</i>: 3.31</td>
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<td>Determined based on linear fitting to the time-series fluorescence measurements from YFP/P2-P3-P1 fusion, as reported in <cite ref="Cheung2008"></cite> and fluorescence per molecule from <cite ref="Wu2005"></cite></td>
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<td>Chung, 2006, and Reshes, 2008</td>
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Revision as of 02:49, 18 October 2014

Math Book: Conjugation

Introduction and Motivation

In order to suppress transcription of the mecA gene, we need a way to deliver our silencing system to the antibiotic resistant cells. To accomplish this, the CRISPRi and sRNA genes are cloned into an engineered conjugative plasmid in closely related cells. The cells with the engineered plasmid are referred to as “donor” cells, and they are introduced to the target population, also known as the “recipients”. The plasmid will transfer from donors to recipients via conjugation, and we refer to these recipient cells with the conjugative plasmid as “transconjugants”. The transconjugants will, in time, be able to retransmit this plasmid. Thus, the system will propagate throughout the infection, thereby disarming the antibiotic resistant cells.

(fig. 1 - diagram of transmission and retransmission of plasmid)

We model the propagation of our plasmid for a number of reasons. Most importantly, we want to determine the optimal time to apply antibiotics to the infection. To do so, we track the total number of donor, recipient, and transconjugant cells and define the “fall time” as the time it takes the number of recipients to reach 10% its initial value. In addition to determining when to apply methicillin, the model can be used to find how large of a conjugation rate is needed and what initial concentration of donor cells is needed to spread the plasmid at a fast enough rate.

Unfortunately, modeling such a system poses quite a bit of difficulty. The primary obstacle is that we consider S. aureus growth on a solid surface (e.g. a lab plate, or on your skin) and not in a well mixed environment. This forces us to abandon more traditional models of conjugation (such as from Levin, Stewart, and Rice) and develop a spatial model instead. We took two main approaches to developing such a model: an agent-based model (ABM), and a partial differential equation (PDE) model.

The ABM is a stochastic simulation of conjugation through a population of donor and recipient cells. The main advantages of this model are that it accounts for randomness of plasmid transfer, as well as likely maintaining accuracy on small-scale areas. Its main drawback is that it is computationally expensive to model a large number of cells, and thus cannot model populations on large scales.

The PDE Model addresses large scale populations through deterministic methods. Being a differential equation system, a wide variety of mathematical tools to analyze the system are readily available, and it is more computationally efficient than the Agent-Based Model.

Agent-Based Model

Overview

We have developed a novel model for bacterial conjugation on flat surfaces. By treating each bacterium as an “agent” that has properties associated with its type (either Donor or Recipient), and letting the group of agents interact over a prescribed period of time, one can make qualitative and quantitative conclusions about their behavior.

The model is based on the idea of having hexagonal cells that each may or may not be occupied by a donor (e.g. modified S. epidermidis) or a recipient (e.g. MRSA). An individual bacterium may divide into an empty neighboring cell and if the cell is a donor, conjugate with an adjacent recipient. We assume that the conjugative plasmid represses methicillin-resistance 100% (i.e. all donor cells will die upon introduction of antibiotic).

The hexagonal grid was used over a traditional square grid because hexagons offer more routes for conjugation and division to occur, as well as the fact that hexagons form a tighter packing structure than squares. Because bacteria are capable of very tight packing on flat surfaces, the hexagonal framework more accurately captures interaction between neighboring bacteria - an important consideration when modeling conjugation.

Algorithm

Before outlining our algorithm, it is important to explicitly state the general assumptions we made in the ABM:

  • The flat surface is treated as a 2D hexagonal grid (of size n x n), and refer to each grid location as a hex
  • Each hex may be occupied by at most one bacterium
  • We do not distinguish between transconjugants original donors
  • Donors and Recipients are both able to divide
  • Donors may conjugate with recipients, after which the recipient becomes a donor
  • A cell may divide if there is an empty adjacent hex
  • A donor may conjugate if there is a recipient in an adjacent hex
  • The natural death rate of cells is negligible (i.e. cells don’t die)
  • All cells are stationary, so population “movement” occurs through division into neighboring cells

The scripts used to run this model were written in Python and follow a very basic structure:

  • Build empty grid
  • Randomly fill with donors, recipients, leaving some percentage of the grid empty
  • Iterate over each grid cell for T units of time:
    • If empty do nothing
    • If recipient, there is a chance to divide if empty adjacent cells exist
    • If donor, there is a chance to divide if not surrounded, and chance to conjugate if there is a neighboring recipient

Flowchart for the Agent-Based Model algorithm

Technical assumptions and details:

  • The algorithm uses weighted uniform distributions (about specified parameters) for conjugation and division
  • Donors lose half of their plasmids when they divide
  • We do not distinguish between transconjugants original donors
  • Cells enter a paused state after they divide or conjugate for a certain amount of time, allowing them to “recharge”

A few potential issues:

  • Donors are given a chance to conjugate before they are given a chance to divide
  • Cells are equally likely to divide, whether it's 100 turns after they're created or 0 turns
  • We iterate through the grid in numerical order, which may lead to an “unnatural” sequence of cell interactions (i.e. inaccuracy). As the simulation “turn time” decreases, the resulting inaccuracy from this behaviour should become negligible
  • The lack of cell death does not seem realistic in a living body

Parameters

Parameter Model Value Experimental Value Comments
Agent division frequency* (Unit: divisions / h) Donor: 0.5, Recipient: 0.5 S. aureus: 1.43, E. coli: 3.31 Chung, 2006, and Reshes, 2008
αmc 0.0011 nM • min-1 mRNA production from Xylose Promoter Same as SarA rate since the addition of the Xylose-inducible promoter was to simplify labwork and thus for modelling we assume it is fully induced.
βc 0.0057-0.4797 protein • transcript-1 min-1 dCas9 protein synthesis rate from dCas9 mRNA Estimated from peptide elongation rates in Streptomyces coelicolor , the dCas9 BioBrick from and ribosome density from report log-phase mRNA half-lives in {S. aureus}. An approximate average value of 4 minutes leads to this degradation rate.
γc, γb -5.6408e-04 min-01 dCas9/complex degradation rate Based off half-life of SarA protein in S. aureus as reported in
Ka 0.28 nM Dissociation constant for complex and DNA (given by k2/k1) found this dissociation rate for dCas9 and a single-stranded DNA substrate.
n 2.5 Hill Constant for Repression UCSF iGEM 2013
k+, k- 0.01 to 1.0 nM Rate of dissociation of dCas9-sgRNA to form complex Range defined relative to other parameters, using the QSSA assumption that these dynamics are fast
Fold Reduction 6 to 35 Maximum percent repression achievable with CRISPRi system Based on the relative fluorescence measurements observed when the non-coding strand was targeted by dCas9 in

Results

PDE Model

Conjugation Images

ABM 0h

Sufficient conjugation rate at t = 0h

ABM 6h

Sufficient conjugation rate at t = 6h

ABM 12h

Sufficient conjugation rate at t = 12h

ABM 18h

Sufficient conjugation rate at t = 18h

ABM 24h

Sufficient conjugation rate at t = 24h

ABM Plot n=10

Sufficient conjugation rate population for 10-by-10 grid

ABM Plot n=100

Sufficient conjugation rate population for 100-by-100 grid

S. aureus conjugation rate at t = 0h

S. aureus conjugation rate at t = 6h

S. aureus conjugation rate at t = 12h

S. aureus conjugation rate at t = 18h

S. aureus conjugation rate at t = 24h

References

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