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ISSN: 2168-9431
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A Self-Consistent Approach to the Lag Phase of Planktonic Microbial Cultures

Schiraldi A*

Department of Chemistry, Division of Physical Chemistry, University of Milan, Milano, Italy

*Corresponding Author:
Schiraldi A
Department of Chemistry
Division of Physical Chemistry
University of Milan, Milano, Italy
Tel: +39 02 503111
E-mail: [email protected]

Received Date: August 07, 2017; Accepted Date: August 11, 2017; Published Date: August 15, 2017

Citation: Schiraldi A (2017) A Self-Consistent Approach to the Lag Phase of Planktonic Microbial Cultures . Single Cell Biol 6: 166. doi:10.4172/2168- 9431.1000166

Copyright: © 2017 Schiraldi A. This is an open-access article 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.

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Abstract

There is no increase of cell density during the lag phase. The cells start duplication with a pace (generation time) that is progressively adjusted during the growth span. This process implies the obvious assumption that the progress of the microbial population via duplication mechanism obeys the law of population density and generation time. This article supports the reliability of the assumption that slope and duration of lag-phase practically counterbalance each other, the small discrepancies being due to the scarcity of the available data and the related uncertainty of estimation, as well as to the fact that the relevant growth process occurred in non-planktonic conditions.

Keywords

Lag phase; Cell density; Population density; Generation time; Planktonic conditions

Introduction

During the lag phase, no increase of cell density takes place. Once ready, the cells start duplication with a pace (generation time) that is progressively adjusted during the growth span. A suitable model [1] to describe this process implies the obvious assumption that the progress of the microbial population via duplication mechanism obeys the law.

Equation (1)

where N0 is the starting population (or population density) and the generation time, τ, is a function of the time, t, namely,

Equation (2)

and b being parameters to asses through a best fit of the plate counts. These parameters reflect the starting growth environment (pH, available volume and substrate, co-existence of other microorganisms, etc.) and its endogenous modifications during the growth span, in the absence of any external perturbation. It comes out that these two parameters may be given a physical meaning [1]: 21/b is the maximum achievable growth extent, namely, Nmax/N0=21/b, 1/b being the average number of duplications for each generation line along the growth path from N0 to Nmax; the ratio (b/a) reflects the sharpness of the growth trend. The maximum specific growth rate, N /N, is attained at t=t*=(a/3b) 1/2 . The tangent to the growth trend at t* crosses the end plateau at tend=3t*.

The reduced quantity, (t)=b log2(N/N0), with Equation can therefore describe the growth extent.

Literature Review

Another peculiarity of a given planktonic culture is the duration of the lag phase, t0, which is not explicitly included in the above model, since it does not imply changes of the population density. Nonetheless, one can easily determine t0 as the shift of the abscissa that allows the compliance with the constraints imposed by the model (Figure 1). The main one is (tend-t0)=3 (t*-t0), having replaced the variable t with (t-t0) in equations (1) and (2), which allows evaluation of t0: (Figure 1).

single-cell-biology-growth-trend

Figure 1: Plot of the growth trend using the reduced quantity ξ that is the achieved fraction of the log2(Nmax/N0) gap. The “origin” of the true growth (dotted line) is identified thanks to the constraint that the tangent at t* must intercept the ξ axis at ξ =-0.125 [1].

Equation (3)

Such a behaviour concerns any microbial species that grows via cell duplication and can be escribed with the reduced quantities: ξ and tR=(t-t0)/(t*-t0). Reminding that (t*-t0)=t*id=(a/3b)1/2, one easily obtains for ξ the expression:

Equation (4)

that does not depend on the parameters a and b and allows one to gather all the growth trends of duplicating microbial species, at any temperature and environmental conditions, in a single master plot (Figure 2).

single-cell-biology-master-plot

Figure 2: The master plot that allows the representation of the growth trend of any duplicating microbial species. The full line corresponds to equation (3), while the data correspond to plate counts reported for different microbes in different environmental conditions and starting from different cell densities. Reproduced from [1].

Self-consistency of the duration of the lag phase

Equation (3) makes the duration of the lag phase consistent with the model. The relationship between the intercept, ξ, and the slope, s, of the tangent at t= t* in the (ξ, t) plan, namely,

Equation

supports this conclusion (Figure 1):

Equation (5)

Equation (5) shows that, for a given γ, a large s (sharp growth trend) corresponds to a small t0, in agreement with most experimental evidences and the prediction of some other models [2]. Now it is possible to draw a new plot (Figure 3) where the time is expressed as (tt 0) /(t*-t0)=tR. There is a definite relationship between |tRo| and γ that comes from equations (4) and (5):

single-cell-biology-master-plot

Figure 3: Master plot of any duplicating planktonic microbial culture (like in Figure 2) that includes the duration of the lag phase according to equation (6). Notice that |γ|≥ 0.

Equation (6)

Figure 3 and equation (6) provide another description of the correlation between the duration of the lag phase and the intercept γ . This suggests a correlation between the relevant biological meanings (Figure 3).

Results and Discussion

Formally, γ reflects a virtual population density, Nstarting, that, starting the growth at t=0, would progress with a fixed duplication rate (dγ/dtR)*=3/8, and reach the levels ξ =-0.125, +0.25 and 1 at tR=0, 1 and 3, respectively, while the actual starting population density, N0, i.e., ξ =0, would be reached at tR=1/3.

Notice that (γ/b) is the average number of duplications for each generation line from Nstarting to N0, namely, Nstarting=N0 2-γ/b. Since the value of (γ/b) is peculiar of the specific culture considered, Nstarting differs from case to case, as well as t0, tRo and (tend-t*).

For the sake of simplifying the picture, one might assume that, for a given microbial species, γ would not change on changing some culture conditions (e.g., temperature), which is tantamount as to assume that s and t0 (equation 5) counterbalance each other, so that (s t0)=constant. One can easily simulate such situation. Keeping b (that is less affected by temperature changes [1]) fixed and varying a, the growth trend would correspond to the picture reported in Figure 4.

single-cell-biology-Growth-trends

Figure 4: Growth trends with a common intercept γ. The used parameters are: b=0.1 and a=1, 2 and 3-time units 2, respectively. The corresponding duration of the lag phase is t0=2.43, 3.44 and 4.22-time units, respectively. The smaller a, the sharper the growth trend.

In another example, a is constant while b changes (Figure 5).

single-cell-biology-Growth-trends

Figure 5: Growth trends with a common intercept γ. The used parameters are: a=5 time units 2, while b=0.1, 0.3 and 1. The larger b, the sharper the growth trend. The corresponding duration of the lag phase is t0=5.44, 3.14 and 1.72 time units, respectively.

Both these examples sketch experimental growth trends. Using the literature data [3] relevant to Pseudomonas fluorescens in a real system, one indeed finds a substantial agreement with the above picture (Figure 6).

single-cell-biology-Growth-trends

Figure 6: Growth trends of Pseudomonas fluorescens in a real system (after [3]), expressed in reduced units as ξ, at various temperatures.

These data support the reliability of the assumption that slope and duration of lag-phase practically counterbalance each other, the small discrepancies being due to the scarcity of the available data and the related uncertainty of estimating γ, as well as to the fact that the relevant growth process occurred in non-planktonic conditions. A mean value of γ comes from the plot of the same data in the (tR, ξ) plan (Figure 7), where a mean value of tR0(-0.34) allows determination of γ (Figure 7).

single-cell-biology-mean-value

Figure 7: The same data as in Figure 6 reported using the reduced time tR. γ is identifying using a mean value for tRo that corresponds to t=0.

Conclusion

Although not explicitly included in the growth model proposed for duplicating species in planktonic conditions [1], the duration of the lag phase, t0, can be easily assessed as the time shift that allows compliance with the growth trend. Once the plate count data are converted in reduced quantities, namely the fraction of the log2 (Nmax/N0) gap, ξ, the slope of the growth trend at t* (where it reached its maximum value) is correlated with t0 and the two quantities almost counterbalance each other. Any experimental growth trend may be supposed to start from a virtual population density, Nstarting<N0, defined by the intercept of the tangent at t*, γ, which does not change too much on changing the culture conditions.

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