Welcome to

metabolator - an online tool for the evaluation of microbial heat flow curves

Introduction

metabolator is an open source software for the quantitative analysis of primarily microbial heat flow curves. It allows uploading data in a predefined format (see below), operates in a semi-supervised fashion and generates tabulated analysis results for download. The user has full control over the data range that is evaluated. If required, a baseline and time offsets can be subtracted. The original data is displayed graphically and the fitted curves superimposed in the same plot. The unique feature of metabolator is the generation of so-called enthalpy plots, i.e., heat flow versus released heat plots. The data are fitted analytically in the enthalpy domain in accordance with the Monod equation and the corresponding time-domain presentation is then generated numerically from the enthalpy-domain fit. The user can save individual fit results using the save option. Saved results can also be cumulated to be finally listed in an Excel table choosing the download option. The table carries the same name as the uploaded dataset with the extension .analysis. In the following, the mathematical background is explained and further details on format requirements are given.

Which information is gained from metabolator

The fit produces three parameters which generate a heat flow curve that approximates the seleced raw data:

H₀: extrapolated total heat release of a metabolic phase (total nutrient consumption), from which a representative fraction has been selected in the enthalpy plot

r₀(m): the maximal growth rate in compliance with the Monod equation 

H_f: the heat release from metabolic activity under the condition that the initial nutrient concentration supported half maximal growth r_i = 0.5·r_0m

Derived parameters:

C_r = H_f / H₀: the relative Monod constant from which the absolute Monod constant is obtained as C_M = C_r·S₀ with S₀ = initial substrate concentration.

r₀(l): the maximal growth rate of logistic growth for the limiting case of H_f and r₀(m) -> infinity.The smaller of the two parameters r₀(m) and r₀(l) is appropriate.

Mathematical background

metabolator describes experimental heat flow curves on the basis of a hyperbolic nutrient dependence of metabolism. This is equivalent to an extension of the Monod equation (ME) beyond the initial exponential phase of a bacterial culture for which the ME originally described the nutrient dependence of the initial exponential growth rate. In contrast, a hyperbolic nutrient dependence of growth is assumed here to be valid also for post-exponential microbial growth, such that in the ideal case, a full “metabolic life span” of a culture can be modeled. This assumes that growth and metabolism stay coupled throughout the “metabolic life span”, i.e., cells continue growing and dividing in proportion to nutrient consumption. Previous work has shown that this is a valid approximation for many microbial cultures. Such modeling is not possible analytically in the time domain, despite the simplicity of the ME. Therefore, metabolator transforms time-dependent data into enthalpy domain, i.e., a plot of the original time-dependent heat flow data P(t) against their integral (the released enthalpy, H). Thereby, the hyperbolic nutrient dependence of microbial metabolism can be treated analytically using the “extended calorimetric Monod equation” (ECME):

P(H) = H·r₀·((H₀−H)/(H₀−H+H_f))   Eq. 1

with H: released heat, H₀: total heat released upon complete nutrient consumption, H_f: total heat released by a culture supplemented with a nutrient concentration that supports initially half-maximal metabolic activity (i.e., a calorimetric homolog of the Monod constant C_M), r₀: maximal division rate. The ECME expresses the ME in calorimetric quantities but additionally introduces the multiplication with H. H scales with the amount of biomass formed during growth. Thus, the multiplication with H up-scales the per cell metabolic activity expressed by the calorimetric ME term to the total amount of cells that contribute to the cumulative heat flow of all cells in the culture. The unit-less calorimetric term:

((H₀−H)/(H₀−H+H_f)) = Θ(H)   Eq. 2

defines the “metabolic load” Θ, which adopts values between zero and unity to express to which degree the maximal metabolic activity of a cell is reached in relation to the activity at saturating nutrient supply. Here, Θ(H) is given by the ME. The ECME expresses the initial growth rates r_i of a metabolic phase as:

r_i = (dp/dH)(0) = r₀·(H₀/(H₀+H_f))   Eq. 3

The total heat release H₀ corresponds to the distance between the two “foot points” of the inverted parabola P(H) on the x-axis of the enthalpy plots generated by metabolator. Hf determines the asymmetry of the enthalpy plot P(H). The value H_f/H₀ =: C_r defines a relative Monod constant, from which the absolute Monod constant is obtained as C_M = [nutrient]∙C_r. If metabolism approaches a linear nutrient dependence, both H_f and r₀ become exceedingly large, which is an intrinsic property of the ME. However, the ratio S := H₀·r₀/H_f = −(dp/dH)(H₀) still has a defined value and determines the sensitivity of metabolic activity to nutrient at vanishingly small concentration (in analogy to the specificity constant r₀/K_M of an enzyme in the Michaelis-Menten model). In such cases, the growth follows a logistic function (Verhulst model) with the maximal growth rate r_0l = 4P_m/H₀, where P_m is the heat flow maximum of the fitted curve.

The strength of metabolator analyses lies predominantly in the determination of H₀ values and of the initial growth rates r_i even in cases, where successive metabolic phases are analyzed. Only the first metabolic phase gives direct experimental access to initial exponential growth. Already the total heat H₀ would be experimentally only directly accessible if all nutrients were consumed in a single metabolic state which is typically not the case. The ECME extrapolates the heat flow curves forward in time to determine the H₀ even for an “incomplete” metabolic phase which is followed by a different metabolic phase before nutrients have been actually fully consumed in the previous metabolic phase. Likewise, the ECME extrapolates the growth behavior back in time, such that  initial growth rates ri can be derived even for late metabolic states which do not contribute to the experimentally observed initial culture growth. This back extrapolation also renders initial growth rate determinations largely independent of very common baseline drifts in IMC data at early growth after insertion of ampoules into the calorimeter. The Monod constants of all sufficiently distinct metabolic phases can be determined from the heat flow curve of a single batch culture.

The example files show these cases.

Using metabolator

metabolator fits an ECME to the data in a range which is set by the user in the “Enthalpy Plot”. The latter can be displayed on the screen as well as the original “Time Plot”. A larger or smaller enthalpy range can be chosen, depending on how well the heat flow of a microbial culture can be described by the ECME. Choosing the lower and upper limit of the data range most critically affects the results and typically requires iterative changes. The program offers the “estimate heat range” option in order to define a plausible initial data range.  This function applies exclusively to the region around the global maximum of a heat flow curve. Shoulders can be fitted as well but require manual setting of the appropriate heat range.

Ideally, the heat flow data should be baseline-corrected and devoid of negative hat flow values. If required, both a “Time offset” and a “Power offset” can be added to the data, such that metabolator uses a corrected data set.

Formats

Input data need to be loaded as excel sheets containing four header lines for sample names, data description etc. The first data column must be time in seconds. All other columns must contain heat flow data in Watt. Typical values are in the 10⁻⁶ W range and must be provided as such (not as µW). The example data files in the folder “Examples” comply with the required format. To save computation time, heat flow curves should not exceed 4000 data points.

Output data contain the fit parameters and statistical information. These are included in an excel file containing the fit parameters in a single row for each data set for which the save button has been activated after the fit procedure. Activating the download field generates an excel file which carries the root name of the original data set with the suffix “_analysis”. H₀ and H_f values are in Joule, r₀ is in h⁻¹. The analysis files further report the released heat H_max at which the maximal heat flow P_max is reached according to the fitted ECME. The additional parameter r₀(l) = 4P_m/H₀ is of interest for those cases, where the ME requires unrealistically large values of H_f and r₀. This is a natural property of the ME for close to linear dependencies of metabolic activity on nutrient, because it is the only parametrization of a hyperbola which can approximate such a linear relation. In these cases growth follows a logistic curve dN/dt = N·k·(1-(N/K)) with a maximal rate k = r₀(l). This value is reported in addition to the maximal rate r₀(m) obtained for the Mono-type of growth. 

User Note

As metabolator is a free research tool under development, you are encouraged to report problems or make suggestions for improvement by mailing to metabolator@hzdr.de.

Your input is greatly appreciated and help can be provided, where problems arise!

K. Fahmy