1 Our collaborations
- 1.1 Modeling for Toulouse 2016
- 1.2 Data recovery Bordeaux 2016
2 Colaboration made by Pretoria 2016
- 2.1 ★ ALERT!
  - 2.1.1 Which other teams can we work with?

Collaborations

Our collaborations

Modeling for Toulouse 2016

Introduction

We conceived a model in order to handle questions concerning the following situation: A bacterial growth is carried out in a bioreactor, continually supplied in substrate. Bacteria can possess 2 type of plasmids:

plasmids 1 carry the toxin A gene and the anti-toxin B gene
plasmids 2 carry the toxin B gene and the anti-toxin A gene

So during the growth each bacterium can have no plasmids, either one type of plasmids, or both types.

The aim of the model is to assess the evolution of plasmids throughout the culture, to determine which parameters can matter in the loss of those plasmids, and to precise what are the probabilities for a bacteria to loose its plasmids during cell division. As a plasmids can be a disadvantage for growth (energy spent into replicating processes) or a advantage (protection against a toxin) this question is hard to answer. But in this situation, where one type of plasmid can influence on the presence of the other type of plasmid in (and reciprocally) in the same bacteria, the question become too tough to answer and only a mathematical model can resolve such a interrogation!

Equations

In 1967 Fredrickson et al. studied mathematically development of a bacterialpopulation, under the assumptions of a large population of independant bacteriain a well mixed solution of constant volume. The large population ensures thatfor the population the expectation value is a good estimate of the average.The bacteria being independant ensures that the behaviour of each individualdepends only on its internal state z and the conditions c which are the samefor all individuals. The volume is well mixed so the conditions c which are thesame everywhere. The volume is constant so that the population caracteristicscan be evaulated by integration over the volume.

From this starting point they develop a pair of master equations of change to describe the evolution of the population:

\frac{\partial}{\partial t} W_\mathbf{z} (\mathbf{z},t) + \nabla_\mathbf{z}\cdot[(\beta\cdot\overline{\mathbf{R}}(\mathbf{z},c)W_\mathbf{z}(\mathbf{z},t))] \\ \\ = 2 \int \sigma (\mathbf{z',c}) p(\mathbf{z,z',c}) W_{\mathbf{Z}}(\mathbf{z'},t)\mathrm{d}v' - (D+\sigma (\mathbf{z,c})) W_{\mathbf{Z}}(\mathbf{z},t)

(1)

\\ \\

\frac{d\mathbf{c}}{dt} = D(\mathbf{c_f} - \mathbf{c} ) + \mathbf{\gamma}\cdotp \int \bar{\mathbf{R}}(\mathbf{z,c}) W_{\mathbf{Z}}(\mathbf{z},t)\mathrm{d}v

(2)

In these equations the various symbols are as follows:

\mathbf{z}	Vector for internal state of a bacteria.
\mathbf{c}	Time dependant vector for conditions.
W_\mathbf{z} (\mathbf{z},t)	Distribution of bacteria in z, t space.
\overline{\mathbf{R}}(\mathbf{z},c)	The expected value or the reaction rate vector in z, t space.
\sigma (\mathbf{z,c})	Rate of fision for bacteria in z, c space.
p(\mathbf{z,z',c}))	Partitioning probability of generating a child in state z from a parent in state z'.
\nabla_\mathbf{z}\cdot\mathbf{V}	\sum \frac{\partial}{\partial z_i}\mathbf{V}_i
\mathrm{d}v'	Integral over state space v'
D	Dilution rate of the culture (for femrenters).
\beta	Stochiometric matrix for cellular substances.
\gamma	Stochiometric matrix for extra-cellular substances.

With these relations:

\dot{\mathbf{V}}(\mathbf{z,c}) = \mathbf{\beta} \cdotp \mathbf{R}(\mathbf{z,c})

The expected internal state change rate vector.

\\ -\mathbf{\gamma} \cdotp \bar{\mathbf{R}}(\mathbf{z,c})

The expected consumation of substances in the environment by a cell in state z.
Thus for a particular problem in hand it is necessary to chose z and c that represent the state of cells and the media. Then the matrices and functions

\beta, \gamma, \mathbf{R}(\mathbf{z,c}), \sigma (\mathbf{z',c}) and p(\mathbf{z,z',c})

need to be defined for the problem considered. Finally the inital conditions

W_{\mathbf{Z}}(\mathbf{z},t)

and

c_0

and growth conditions D and

c_f

need to be fixed.

For the problem in hand, plasmid maintenance during growth with 2 different plasmids, and attempting to find a simple solution to the problem we propose a 3 variable internal state vector:

$$\mathbf{z} = \begin{bmatrix} z_0 \\ z_1 \\ z_2 \end{bmatrix} = \begin{bmatrix}\textrm{Cell maturity} \\ \textrm{count of plasmid 1} \\ \textrm{count of plasmid 2} \end{bmatrix}$$ \\

In this internal state vector

is a mesure of the growth of the bacteria,encompassing such things as size, number of chromosomes and mass,

and

represent the number of copies of each plasmid. For the external conditions wepropose simply the substrate concentration S. The maturity has a minimumvalue of 1 and must increase to 2 before division can occur.

For the rates of change of the internal state vector then we propose for the bacterial maturity to extend the development presented in Shene et al. [?] to include 2 plasmids and incorporate cell maturity as a state variable. This gives:

\dot{z}_0 (\mathbf{z},S) = \mu = \mu _{max} \frac{S}{K_S+S} \frac{K_{z_1}}{K_{z_1}+z_1^{m_1}} \frac{K_{z_2}}{K_{z_2}+z_2^{m_2}} \\

(3)

Here

\mu _{max}

is the maximum growth rate

hr^{-1}: $\mu (\mathbf{z,S})

the growth rate ;

K_S

is the Monod constant in g/l for the substrate;

K_{z_1}

is the inhibition constant for plamid number 1 in (plasmids per cell)

^{m_1}

, and

m_1

the Hill coefficient for the cooperativity of inhibition.

K_{z_2}

and

m_2

represent the same parameters for plasmid 2. For plasmid replication rate we propose, again following Shene et al. [?], the empirical relationship :

\dot{z}_1 (\mathbf{z},S) = k_1 \frac{\mu (\mathbf{z},S)}{K_1 + \mu (\mathbf{z},S)} ( z_{1_{max}} - z_1 ) if z_1 \geq 1.0 or \dot{z}_1 (\mathbf{z},S) = 0 if 0.0 \leq z_1 < 1.0

(4)

This relation, and equivalent one for plasmid number 2

\dot{z}_2 (\mathbf{z,S})

is designed to satisfy the boundary conditions of no reproduction if there is less than 1 plasmid in the cell, and a maximum copy number of

z_{1_{max}}

. This introduces the parameters

k_1

and

K_1

which are respectively the plasmid replication rate (in

hr^{-1}

) and the inhibition constant (also in

hr^{-1}

).
The inhibition constant reduces plasmid replication rate at slower growth rates. Notice that here we have directly defined

\bar{\dot{\mathbf{V}}}(\mathbf{z,c})

rather than

\beta

and

\bar{\mathbf{R}}(\mathbf{z,c})

.
For the growth yield we propose :

\gamma \cdotp \bar{\mathbf{R}}(\mathbf{z,c}) = \alpha \mu(\mathbf{z},S)

(5)

Where alpha is the growth yield in

g/l/cell

. The remaining functions and parameters in equations 1 and 2 are the division rate

\sigma (\mathbf{z,c})

and the partitioning 2 function

p(\mathbf{z,z',c})

. There is less consensus in the litterature for an at least empirically appropriate form for these equations. To remain simple we propose:

\sigma (\mathbf{z,c}) = \sigma \times H[2.0] = 0 if z_0 < 2.0 \\ \sigma (\mathbf{z,c}) = \sigma \times H[2.0] = \sigma if z_0 \geq 2.0

(6)

Here we assume that there is a fixed rate of division

\sigma

once cells are big enough to divide (

H[]

is the Heaviside function).

p(\mathbf{z,z',c}) = p(z_0,z'_0) \times p(z_1,z'_1) \times p(z_2,z'_2)

(7)

p(z_0,z'_0) = \delta_{z_0,\frac{z'_0}{2}} = 1 if z_0 = z'_0/2.0 \\ p(z_0,z'_0) = \delta_{z_0,\frac{z'_0}{2}} = 0 if z_0 \ne z'_0/2.0.

(8)

p(z_1,z'_1) = 0.5^{z'_1} \begin{pmatrix} z_1 \\ z'_1 \\ \end{pmatrix} = 0.5^{z'_1} \frac{z'_1 !}{(z'_1-z_1)! z_1 !}

(9)

In these equations we assume that the partitioning of the three internal state variables are independant. That cells divide exactly in half, that is the maturity parameter is exactly halved when the cells divide (

\delta

is a Kronecker delta function). That the two plasmids segregate independantly and as individual plasmids according to a binomial distribution. These assumptions are probably the most suspect in the model. This initial version of the model has no contention, that is

z_1

and

z_2

do not influence the growth rate

\mu

. In order to develop the model for the system envisaged this needs to be introduced.

Progamming code

Code de françois à ajouter

Results

METTRE LES COURBE

Data recovery Bordeaux 2016

In order to help the iGEM team Bordeaux, we had to read the paper « Dynamics of plasmid transfer on surfaces » and collect and organize some data about their experiments on plasmids transfer. Thanks to this, the iGEM team Bordeaux can compare those results to their own computaionnal results.

Colaboration made by Pretoria 2016

The iGEM team Pretoria 2016 hepled us focusing on the socio-economic and political issues facing the current platinum sector, including the Marikana strikes.

★ ALERT!

This page is used by the judges to evaluate your team for the <a href="https://2016.igem.org/Judging/Medals">team collaboration silver medal criterion</a>.

Delete this box in order to be evaluated for this medal. See more information at <a href="https://2016.igem.org/Judging/Evaluated_Pages/Instructions"> Instructions for Evaluated Pages </a>.

Sharing and collaboration are core values of iGEM. We encourage you to reach out and work with other teams on difficult problems that you can more easily solve together.

Which other teams can we work with?

You can work with any other team in the competition, including software, hardware, high school and other tracks. You can also work with non-iGEM research groups, but they do not count towards the iGEM team collaboration silver medal criterion.

In order to meet the silver medal criteria on helping another team, you must complete this page and detail the nature of your collaboration with another iGEM team.

Here are some suggestions for projects you could work on with other teams:

Improve the function of another team's BioBrick Part or Device
Characterize another team's part
Debug a construct
Model or simulating another team's system
Test another team's software
Help build and test another team's hardware project
Mentor a high-school team

Team:Aix-Marseille/Collaborations

Contents