Team:Toulouse/Modelling

Modeling is a tool used to simplify and study systems. It helps us to predict behavior thanks to bibliographic or experimental informations.
The following modelisation focuses on the development of our engineered bacterium (called SubtiTree) in trees. The bacterial growth in trees seems to be unknown, thus we must infer Bacillus subtilis' behavior.

Bacterial Growth

Aim

Bacillus subtilis is a tree endophyte strain. A study showed that Bacillus subtilis could develop and fully colonize a tree, reaching a concentration of 10⁵ cells per gram of fresh plant. We need to know in which conditions the growth of B. subtilis is optimum in a tree and if the weather can stop its development during winter. Therefore we decided to work on the Bacillus subtilis' growth in function of the temperature during the year.
Modeling bacterial growth in a tree section generates some difficulties. We need to know the distance between two tree extremities (treetops and root) or the speed sap flow. However the speed sap flow can vary with temperature during the day and seasons cause of the type of sap (phloem, xylem). Furthermore a tree is not an homogeneous system: its roots, trunk and branches do not contain the same amount of sap and wood.
The average speed of the plane tree sap is 2.4m/h, which means that in a day the sap of a 30m tree will flow from one extremity to the other. Tree is reduced to a bioreactor.

We make the following hypothesis:

1. According to the publication of Xianling Ji (See References), after six months of Bacillus subtilis growth in a tree, bacteria cells reach a concentration of 10⁵ cells per gram of fresh plant. We assume that 10⁵ cells/g is the maximum concentration.
2. The composition of the phloem is stable. There is no effect of depletion of the medium.
3. Only temperature impacts on bacterial growth.
4. It is assumed that there is no leakage of cells.

Method

An assessment of the Bacillus subtilis growth in a similar sap was performed in laboratory conditions with optimum growth medium for Bacillus subtilis. The composition sap used was the one from birch sap.
In these conditions, the growth rate μ is optimal. From this value we can extrapolate a growth curve as a function of temperature. We used the cardinal temperature model:

T: Temperature
µ_opt: Optimal growth rate
µ: growth rate at temperature T
T_max: Maximum temperature supported by bacteria
T_min: Minimum temperature supported by bacteria
T_opt: Optimum temperature for the growth

Necessary parameters for this function are minimun temperature T_min and maximum temperature T_max, optimal temperature for the growth T_opt and optimal growth rate µ_opt.

T_min: 10°C
T_max: 52°C
T_opt: 37°C
µ_opt: 8.5968 cfu/d

The optimal growth rate (µ_opt) is obtained experimentally with a similar birch sap environment.
The growth rate is negative below 10°C (according to growth tests performed at 10°C and 4°C under similar conditions for the measurement of μ_opt), survival rate after 24h was 0.3 % at 10°C and null at 4°C.
Conditions apply:

If__| T<= 4°C -> µ = -1
____| 4°C µ = -0.97
____| T > 10°C -> µ = f(T) with f(T) egal to cardinal temperature model.

Figure 1: bacterial growth (µ) as a function of temperature

A logistic model developed by Hiroshi Fujikawa (See References) is used to study bacterial growth.

General logistics formulas:

In our case, the growth rate µ depends on the temperature.
N corresponds to the bacterial population, N_min and N_max are two asymptotes.
The parameter m is a curvature parameter. Larger m is, smaller is the curvature of the deceleration phase with the model.
The parameter n is a parameter related to the period lag. Larger n is, shorter is the period of lag.

N_min is slightly lower than N₀. When N is small at the initial state (N = N₀) i.e. N is close to N_min, N_min/N is almost equal to 1. Therefore the term (1-(N_min/N)) is nearly 0 and the growth is very slow.
If N decreases until it reaches N_min, the term (1-(N_min/N)) is equal to 0. Therefore the growth is null.
Similarly when N is equal to N_max, the term (1-(N/N_max)) is equal to 0 and the growth is blocked.
To overcome this, we worked under two conditions: positive and negative growth. Theses conditions can be translated in two equations. This leads to the writing of this model:

with n = 1 and m = 0.5

The term (1-(Nmin/N)) is not taken into account when there is growth.
The term (1-(N/Nmax)) is not taken into account when there is bacterial decay.
Meteorological records of the Toulouse region during the years 2011-2013 are used to do average daily temperatures. Thus we can determine B.subtilis growth in a tree located in Toulouse during a year. This values are obtained for each day by the average on the highest and the lowest temperature.
The density of green wood plane is about 650kg/m³. The average diameter of the trunks of the concerned trees is about 0.80m and 15m high. This represents a volume of 30m³. Therefore the weight of the trunk is 19.604kg. We need to add to this weight the weight of branches, twigs, about 25% of leaves and about 15% of roots (source-FR).
The average weight of a tree plane is 27,446kg. We inoculated 10mL of bacterial culture at 10⁹cfu/mL, i.e. 10^10 bacterial cells. This represents 3.64x10²cfu/g of fresh plant (N0).

Figure 2: (black) Bacillus subtilis growth curve during one year (N is cell quantity by g of fresh plant). (red) average temperature. (blue) threshold at 10°C.

In our model, growth starts only from 10°C, which happens between March and April. This period seems to be suitable to put the strain in the tree. From December the temperature decreases below 4°C corresponding to the threshold below which bacteria die.

Discussion

In practice, temperature variations are certainly lower in trees than outside, especially if roots extend very deep. Composition of the tree sap must also intervene in the growth rate and nutrient content of sap is also temperature dependent. The effects of the decrease of the temperature in winter also induces a fall of the sap and this must also be involved in the disappearance of our strain in the tree. The period of Bacillus subtilis growth is certainly affected by the change of temperature, the rise of sap ans sap composition variations. All these parameters can consequently slow or fast the growth rate. The modeling work is done with the programming language 'R' script attached (See Annexe).

References

• Xianling Ji, Guobing Lu, Yingping Gai, Chengchao Zheng & Zhimei Mu (2008) Biological control against bacterial wilt and colonization of mulberry by an endophytic Bacillus subtilis strain. FEMS Microbiol Ecol 65: 565–573

• A. Garnier(1977) Transfert de sève brute dans le tronc des arbres aspects méthodologiques et physiologiques. Ann. Sci. Foresi. 34 (1): 17-45

• Heikki Kallio , Tuija Teerinen , Seija Ahtonen , Meri Suihko , Reino R. Linko (1989) Composition and properties of birch syrup (Betula pubescens). J. Agric. Food Chem 37 (1): 51–54

• L. Rosso, J. R. Lobry & J. P. Flandrois (1992) AN Unexpected Correlation between Cardinal Temperatures of Microbial Growth Highlighted by a New Model. J. theor. Biol. 162 : 447-463

• Hiroshi Fujikawa (2010), Development of a New Logistic Model for Microbial Growth in Foods. Biocontrol of Science Vol 15: 75-80

Annexe

To upload the script and the table Click Here

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