Analysis

General approach

After demonstrating the feasibility of our synthetic consortium, we needed to understand further its operation, and in particular to characterize some emerging properties that drive its functioning to ultimately improve its efficiency. We have implemented and performed global sensitivity analyses to test the robustness of the system under real life conditions (such as fluctuations of some parameters that can arise from several steps, as detailed below). Then, to optimize its behavior and guide its design, we carried out more detailed analyses by extending the Metabolic control analysis framework classically used in the fields of metabolic engeneering and systems biology.

Global sensitivity analysis

All parameter values cannot be measured experimentally ¹, and our experience in this project, with a model composed by nearly fifty parameters, confirms this fact. As an additional concern, many sources of variability may perturb parameters and ultimately impact the efficiency of the system. Variability could emerge from the system itself (living systems exert some variability by nature) or during each step of the process (manufacturing, transport, storage, etc). Facing this problem, a global sensitivity analysis approach was applied to evaluate the impact of random parameters fluctuations on the response time to reach non-pathogenic concentrations.¹

To perform this analysis, we generated 10,000 sets of parameters randomly sampled within ± 10% of their reference values, using a uniform distribution, and the response time of the system was calculated for each of these sets (Matlab code available below). Simulation results were analyzed with RStudio (R code available below).

Global analysis files: Global_Analysis.m + System_of_ODEs.m + Resolution_Function.m + myEventsFcn.m + Global_analysis.R

Distribution of the response time
for 10,000 sets of random parameters

Simulation results indicate that the response times varied between 45.46 min (minimum) and 65.05 min (maximum), with a median of 53.70 min and a mean of 53.87 min close to the response time of the initial set of parameters (53.6 min). The standard deviation of the response time was low (±3 min).

These statistical results confirm the global robustness of the system, which will remain efficient even under a reasonable degree of uncertainty.

Metabolic control analysis

Metabolic Control Analysis (MCA) is a mathematical tool widely used in biotechnology to quantify the influence of a specific parameter on the functioning of the system, in terms of fluxes and concentrations. Working with a system governed by a large number of parameters, MCA allows to determine the influence of each of them towards each variable of the system.²

Usually applied to investigate the control of concentrations and fluxes under steady state conditions, we have extended the concepts of MCA to quantify the control exerted by each parameter on the response time (τ), which depends on the system dynamics. For each parameter (p), the control coefficient (C) was calculated as:

\begin{equation*} C = \frac{(\tau(p)-\tau((1+\delta).p))/\tau(p)}{(p - (1+\delta).p)/p} \end{equation*}

Each coefficient quantifies the relative change in the response time τ in response to a relative change δ of the parameter p. If the response time is not impacted by the parameter p, the corresponding coefficient will be zero. A positive (negative) value indicates that an increase in p increases (reduces) the response time. A coefficient of 1 indicate that a change in x % of the parameter results in a change of x % of the response time.

Control coefficients were calculated by numerical differentiation (δ set at 0.01). To reduce calculation times and increase the accuracy of the calculated coefficients, we used the event function of Matlab: integration stops as soon as the concentration in Vibrio cholerae reach a non-pathogenic concentration, and the solver returns the exact τ value. This sensitivity analysis was performed for a wide range of initial concentration of Vibrio cholerae to ensure the genericity of its outcome, and results are presented as a heatmap. The Matlab script used to calculated control coefficient and generate the heatmap (SimBiology needed) are available below.

MCA files: MCA_ev.m + myEventsFcn.m + System_of_ODEs.m

Control coefficients of each parameter on the response time, calculated for different concentrations of *V. cholerae*. Enlarge

Red: parameters favouring a long response time
Green: parameters favouring a short response time
Black: parameters with no significant influence on the response time

Important conclusions can be reached from this analysis. First, we notice that the initial concentration in Vibrio harveyi as well as most of its parameters have a weak influence on the response time (coefficients close to 0, white). This implies that V. harveyi works as a simple inducer: a small number of cells is sufficient to sense CAI-1 and produce enough diacetyl to activate Pichia pastoris. On the contrary, Pichia pastoris concentration and other parameters related to AMP efficiency (IC50, k kill Vc) exert a significant control on the response time, because a high antimicrobial peptides (AMP) concentration is required to kill V. cholerae. Efforts to optimize the system and speed up its response should thus focus in engineering the expression system (P. pastoris) to increase the rate of AMPs production, or to improve the efficiency of AMPs (e.g. by using a different AMP), rather than improving the sensor and transmission parts. Interestingly, the device volume also appears to be a key parameters for our system. For further developments, we should consider carefully AMP properties (death rate, IC50 regarding Vibrio cholerae), as well as the AMP expression system Pichia pastoris and its concentration in the device.

Conclusion

Our model analysis gave us promising results about the efficiency and robustness of the system. We demonstrated it shows a robust behavior under fluctuating conditions, and identified the most controlling parameters that should be tuned to improve its response, hence guiding the design of an improved system. Pichia pastoris initial concentration ([Pichia pastoris]₀) appeared to be the more relevant parameter to optimize, because its value can be directly changed without any molecular or microbial engineering approach. To evaluate further its impact, we thus simulated the response time for a wide range of P. pastoris concentrations (from 10⁶ to 10¹² cells), and under a wide range of contamination levels (V. cholerae concentration from 4.10⁴ to 4.10⁸).

Response time as function of two major parameters: Vibrio cholerae and Pichia pastoris concentrations

Under all the situations tested here, the response time remains below 120 minutes, even at low Pichia pastoris initial concentrations and very high V. cholerae levels.

This two parameters seemed to be the two more meaningfull to implement in an user-friendly web interface that can be used by non-mathematicians to play with our system.

→ Our visual interface is available on our interface page

References

Kent E, Neumann S, Kummer U, Mendes P, What can we learn from global sensitivity analysis of biochemical systems? PLoS ONE, 8 (2013), p. e79244
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0079244
Moreno-Sánchez R, Saavedra E, Rodríguez-Enríquez S, Olín-Sandoval V, Metabolic control analysis a tool for designing strategies to manipulate metabolic pathways. J. Biomed. Biotechnol., 2008 (2008), p. 597913
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2447884/
Huq A., West, P. A., Small, E. B., Huq, M. I., and Colwell, R. R., Influence of water temperature, salinity and pH on survival and growth of toxigenic Vibrio cholerae serovar O1 associated with live copepods in laboratory microcosms, Appl. Environ. Microbiol. 1984, 48: 420–424.
https://www.ncbi.nlm.nih.gov/pubmed/6486784

Model pages
Overview	Simulation	Analysis	Interface

Team:INSA-UPS France/Model/Analysis