Feb 06, 2020
Version 2
Bioflux Analyses: Modelling V.2
- Tobias Weise1,
- Bettina Boettcher2,
- Slavena Vylkova2
- 1BioControl Jena GmbH, Jena, Germany;
- 2Septomics Research Center, Friedrich Schiller University and Leibniz Institute for Natural Product Research and Infection Biology – Hans Knöll Institute, Jena, Germany
Protocol Citation: Tobias Weise, Bettina Boettcher, Slavena Vylkova 2020. Bioflux Analyses: Modelling. protocols.io https://dx.doi.org/10.17504/protocols.io.bb7sirne
Manuscript citation:
This work was supported by the German Ministry for Education and Science in the program Unternehmen Region (BMBF 03Z2JN11).
License: This is an open access protocol distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited
Protocol status: Working
We use this protocol and it's working
Created: February 05, 2020
Last Modified: February 06, 2020
Protocol Integer ID: 32722
Keywords: Candida Albicans, Biofilm, Biofilm Formation, Growth Rate, ODE Model, Parameter Estimation
Abstract
Biofilm formation under shear flow conditions was monitored using the Bioflux1000 device (Fluxion Biosciences, Inc.). In short, Candida albicans overnight cultures were washed in pre-warmed RPMI medium. Cells were seeded for 2-5 sec from the outlet well into the channels of Bioflux1000 flow chambers, which were primed before with warm medium. The cells were allowed to adhere to the channels for 90 min without any flow, followed by removal of non-adherent cells by flowing fresh, pre-warmed RPMI medium for 5 sec. Shear flow was set for time series experiments over 24 h biofilm formation and images were captured every 20 min. Two channels were investigated in parallel having a 10 × magnification to allow a direct comparison between a mutant and a reference (wild-type) strain. Image capturing and stacks to movies was performed using the MetaMorph® Software (Molecular Devices).
An ODE model reflecting the logistic growth as well as the lag phase was fitted to the individual experiments. Fitting was carried out by minimising a cost function (unweighted least-squares-based) using the Nelder-Mead algorithm. Growth rate time series generated from the fitted model were used to compare wild type and mutant regarding the maximum observed growth rates at their respective time points.
All computations were performed using the programming language python (version 3.6.9) and the additional packages numpy (version 1.16.2), pandas (version 0.25.0), and scipy (version 1.3.1).
Modelling Approach
Modelling Approach
The modelling presented here uses a simple ODE model fitted to the individual Bioflux Ipix time series data sets (Fig. 1-A) in order to estimate the height as well as the time point of maximum observable growth (Fig. 2-B).
Fig. 1. Example of Bioflux time series data sets; A: medium pixel intensity; B: calculated growth rate.
Table 1 contains symbols and abbreviations used within the modelling approach and parameter estimation.
Table 1. Symbols and abbreviations used within the modelling approach and parameter estimation.
Ordinary differential equation for discontinuous bioprocesses
The modelling presented here was carried out using the ordinary differential equation for discontinuous bioprocesses with respect to the mean pixel intensity calculated according to the protocol 'Bioflux Analyses: Image Preprocessing'.
Growth Kinetics
The growth of the culture was described using the following kinetics:
- logistic equation (Verhulst-eqation; employed to modell the capacitive limit of the biomass accumulation)
- lag phase (Wolf-equation; employed to modell the initial lag phase of the culture due to adaptation)
The model parameters µmax, Ipix,max, ti as well as the start condition Ipix,0 is estimated within Step No. 2. The adherence time td represents the time difference between the seeding of the cells an the start of the experiment/simulation. td was set to 1.5 h for all simulations in accordance to the experimental set-up.
Import of the required packages.
import numpy as np
import pandas as pd
import scipy.integrate
from scipy.optimize import fmin
Definition of the growth kinetics.
# logistic equation (Verhulst)
def chi(Ipixmax,Ipix):
return 1 - Ipix / Ipixmax
# description of lag phase (Wolf-term)
def wolf(ti,td,t):
return 1 - np.exp( - ( t + td ) / ( ti + td ) )
# growth kinetics
def mu(mumax,Ipixmax,ti,td,Ipix,t):
return mumax * chi(Ipixmax,Ipix) * wolf(ti,td,t)
Definition of the ODE model.
# Model
#===========================================================================
def eq(indep_par,exp_par,start_t,end_t,incr):
initial_cond = indep_par[0],0
mumax,Ipixmax,ti = indep_par[1:6]
td = exp_par
t = np.linspace(start_t, end_t,incr) # time grid
# ------------------------------------------------------------------------
def funct(y,t): # ODE system
Ipix = y[0] # biomass
tin = y[1] # internal t (needed for Wolf-term)
dIpixdt = mu(mumax,Ipixmax,ti,td,Ipix,tin) * Ipix
dtindt = 1
return [dIpixdt,dtindt]
# ------------------------------------------------------------------------
ds = scipy.integrate.odeint(funct,initial_cond,t) #integrate
return (ds[:,0],t)
Parameter Estimation
Parameter Estimation
Based on the ordinary differential equation above, the experimental Ipix time series data sets were simulated individually for the respective experiment. The time series were simulated for an experiment time of 24 h. Numerical integration was carried out employing lsoda from ODEPACK (scipy.integrate.odeint).
The objective of the parameter estimation was to identify the parameter values for the respective experiment. The model parameters µmax, Ipix,max, ti as well as the start condition Ipix,0 are vectorised as θ. The optimum parameter values Ω were estimated by minimising the cost function J(θ) using the downhill-simplex method (Nelder-Mead method; scipy.optimize.fmin). Standard settings regarding the convergence criteria were used with respect to the above package.
Definition of the scoring function (least-squares).
# definition of scoring function (least-squares beased)
def score(parms):
# score difference between model and data points
ss = lambda data,model:(( data - model )**2 ).sum()
# get solution to system
Ipix_model,t_model = eq(parms,exp_rates,0,time,100)
# model index to compare to data
mt=np.linspace(0,time,100)
findindex=lambda x:np.where(mt>=x)[0][0]
Ipix_index=list(map(findindex,data.time))
data.Ipix_model = Ipix_model[Ipix_index]
return ss(data.Ipix,data.Ipix_model)
Execution of the parameter estimation.
initials = [Ipix_zero,mumax,ti,Ipix_max]
exp_rates = 1.5 # [h]
time = 24 # [h]
answ = scipy.optimize.fmin(score, rates, full_output=1, retall=1, maxiter=1000)
estimates = answ[0]
score_value = answ[1]
Based on the parametrised model Ipix and growth rate time series were simulated for each experiment (Fig. 2-A and B). The subsequent evaluations were carried out using the modelled time series.
Fig. 2. Example of the fitted Bioflux time series data sets; A: medium pixel intensity; B: growth rate.
According to Fig. 2-B, the height as well as the time point of the maximum observable growth rate is recorded. These values are used for the comparison between the wild type an their respective mutant.
Case Study
Case Study
In order to illustrate the differences in the biofilm formation between wild type and mutant strains, the observed maximum growth rates are plotted against their respective time points (Fig. 3.1)-A. The differential plot (Fi. 3.1-B), conferring to subfigure A, presents these differences between the mutant strains (arrow head) and their respective wild types (arrow tail).
Fig. 3.1 Maximum observable growth rates max(µ) at their respective time points for wild type and mutant strains; A: Scatter plot of max(µ) at time point t from two examples (wild type vs. mutant); B: Difference plot for max(µ) at time point t, wild type (arrow tail) to respective mutant (arrow head).