Biotechnol. Pmg. 1003, 9, 109-112
A Correlation for the Effective Strength of Escherichia coli during Homogenization Anton P. J. Middelberg' and Brian K. O'Neill Co-operative Research Centre for Tissue Growth and Repair, Department of Chemical Engineering, The University of Adelaide, G.P.O. Box 498, Adelaide, South Australia, 5001,Australia
A new model for high-pressure homogenization has been previously developed. A key model parameter, the mean effective cell strength, can be correlated with average cell length and peptidoglycan cross-linkage. In this article, we develop a correlation for mean effective strength based on a statistical thermodynamic approach to fracture. The final correlation provides an unbiased estimate. While it offers no numerical advantage over a previous empirical correlation, it is based on a modeling approach and an understanding of wall structure. The variable groupings are therefore justifiable.
Introduction
A new model for the disruption of Escherichia coli by high-pressure homogenization has been presented (Middelberg et al., 1992a). Briefly, cells are assumed to have a distribution of effective strengths, fs(S). An individual cell experiences a disruptive force during homogenization and will be destroyed if this exceeds ita effective strength. The fraction of disrupted cells (of effective strength S) is given by (1) U S ) = fs(S) f D W d S where f ~ ( sis) the fraction of events which result in a disruptive force greater than S. Integration gives the total disruption for a population.
A bimodal normal distribution for effective strength is employed. This is justified as a given E. coli population will consist of dividing and nondividing cells. The division site may act as a stress-concentration point, so that septated cells may be weaker. The distribution of effective strengths for a cell population is given by
where x , is the volume fraction of the population which is septated. Subscripts s and n denote the septated and nonseptated subpopulations, respectively. The homogenizer stress distribution, f D ( s ) , is given by (4)
where m, n, and d are system-specific and cultureindependent constants and P is the homogenizer pressure. Equation 4 is based on studies of small-cylinder impact against a plane surface (Vervoorn and Austin, 1990). The values of eight parameters are required to calculate disruption. Regression studies of experimental disruption
* Corresponding author. 8756-7938/93/3009-0109$04.00/0
data for wild-type E. coli B using an APV-Gaulin 15M8TA homogenizer showed that six parameters (Sa,us)un, m, n, d ) were approximately constant and equal to the values given in Table I (Middelberg et al., 1992a). A total of 21 cultures exposed to different periods of glucose exhaustion were examined for disruption pressures varying up to 75 MPa. A total of 182 data points were used in the regression. For a different strain of E. coli or a different homogenizer, the values of the six constants may be expected to change. In particular, the three parameters in eq 4 are expected to depend on the system employed. To use the model in a predictive capacity, values for the remaining two parameters ( x , and Sn) are required. The septated volume fraction, x,, can be measured directly (e.g., by image analysis). The mean effective strength of nonseptated cells, S,, can be correlated with measurable cell properties. Equation 5 was determined for the previous experimental study:
Sn= 3 4 . 2 -~ 8.61Ln ~ + 48.2
(5)
where x , is the degree of peptidoglycan cross-linkage and
L n is the average length of nonseptated cells (Middelberg et al., 1992b). Both independent variables were measured experimentally. Peptidoglycan cross-linkage was determined using high-performance liquid chromatography. Average length was determined by image analysis. Mean effective strength, and hence disruption, may therefore be predicted using measurable culture characteristics, provided the system- (m,n,and d ) and strain-specific (S,, us, and a,) constants are known. Equation 5 gives a good prediction of mean effective cell strength (coefficient of determination 0.938). However, ita form is entirely empirical. The relationship between mean effective strength and culture characteristics cannot be justified from an understanding of cellular structure. In this article, we briefly review cell-wall structure and develop a correlation for mean effective strength based on a statistical thermodynamic approach to fracture. The final correlation is based on a physical understanding of the effect which the key measured variables ( x , and L,) have on strength.
Structure of the E. coli Cell Wall The simplified structure of E. coli peptidoglycan is shown in Figure 1. Glycan chains composed of N-acetylglucosamine (NAG) and N-acetylmuramic acid (NAM)
0 1993 American Chemical Society and American Institute of Chemical Engineers
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60%
1 j,
0
0
50%
m
0
8
0
3
fi 0
0
x
4
2 40% 3 Ee, Y
-I
I
I
NAM
*
Two NAM units
crosslinked by peptide bond TwoNAMunits without crosslink
30%
1
Figure 1. Idealized structure of a single peptidoglycan layer. Table I. Parameter Values for Equations 3 and 4 parameter value parameter value m n
13.0 0.380 7.41
d
"
an
s,
1.71 3.17 17.71
are cross-linked together through peptide bonds. There is evidence to suggest that glycan chains are aligned perpendicular to the main (or long) axis of the bacterium (Verwer et al., 1978). Early models proposed peptidoglycan to be a single layer. Subsequent evidence has shown that it is actually multilayered (Glauner et al., 1988) and that its thickness increases from 6.6 f 1.5 to 8.8 f 1.8 nm during the transition from exponential to stationary phase (Leduc et al., 1989). This corresponds to 2-3 and 4-5 layers of peptidoglycan, respectively. Figure 2 shows the relationship between peptidoglycan cross-linkage, x,, average cell length L,, and mean effective strength, S , (data from Middelberg et al. (1992b)). Cell strengthening initially occurs through a reduction in average length at constant cross-linkage. Mean effective strength increases by approximately 35 % at constant crosslinkage (from 31 to 42). The increase in peptidoglycan thickness during the transition to the stationary phase is approximately 33% (from 6.6 to 8.8 nm), ignoring the experimental uncertainty. The initial strengthening may therefore be due to an increase in peptidoglycan thickness. This increase in thickness may be accompanied by a reduction in averagecell length. In the second phase, mean effective strength increases through an increase in peptidoglycan cross-linkage.
Development of the Correlation Figure 1 shows the simplified structure of a single peptidoglycan layer. A series of harmonic springs (peptide bonds) are arranged in parallel and subjected to a common stress. The free energy density of such a structure at fixed stress is given by (Blumberg Selinger et al., 1991)
35
30
40
45
50
55
Mean Effective Cell Strength Figure 2. Relationship between peptidoglycan cross-linkage, average length of nonseptated cells, and the mean effective cell strength 0 ,total murein cross-linkage;0,average length (wm).
(Blumberg Selinger et al., 1991). The global minimum of the free energy function is the fracture state of all bonds broken ( x = 0). For low applied stress, u, the function exhibits a local minimum indicating a metastable state. The local and global minima are separated by a specific free energy barrier, which decreases as u is increased (Blumberg Selinger et al., 1991). The derivatives ofg with respect to x are the following: ag -=--
ax
q2
2KX2
33 + T[ln x -In (1- x ) ]
(7)
A t a certain limiting stress level, UL, the free energy barrier will disappear. A plot of g versus x will exhibit a stationary inflection located at some point XL. This point, and the corresponding stress level, can be found by setting eqs 7 and 8 equal to zero and solving for XL and a. Specifically, XL can be determined by solving eq 9.
Consider a single layer of peptidoglycan with cross-linkage x,. If 2, is greater than XL, the layer will not fracture until UL is exceeded. For x , less than XL, the layer will disrupt before UL is reached. When the maximum of the free energy barrier occurs at x,, the cell is at its stability limit. A further stress increase will shift the free energy barrier past x,. The cell will disrupt. The critical stress is obtained by setting eq 7 equal to zero. The final equations for the critical stresswhich a single layer can support are therefore the following:
(loa)
2
+ T[x In x + (1- x ) In (1- x ) ] (6) is the fraction of intact bonds, a is the bond U
g(x,u) = - -- ax
2KX
where x dissociation energy, K is the bond elastic modulus, and T is the temperature. These variables have arbitrary units
With the first derivative set equal to zero, eq 8 may be
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55
.s M 8 50 b
v1 e,
.->
t( 45
g
1
40
'c)
Bu 3 2 a
35
30 35
40
45
50
55
Mean Effective Strength Figure 3. Parity plot comparing mean effective cell strength with predicted values from eqs 5 and 14.
expressed as
50%
25%
0%
30
15%
100%
Peptidoglycan Crosslinkage Figure 4. Free energy (eq 6) versus peptidoglycan cross-linkage at various stress levels: -, u = 0.75;- -, u = 1.25; - - -,u = 1.75; - - -, u = 1.93; - - - -, u = 2.25.
sponding critical stress level, a,of 1.93. A parity plot is shown in Figure 3.
s, = -122*67xc L, + 2.763 For x c I X L , the second derivative is always less than zero, and eq 10a guarantees that the maximum of the energy barrier, and not the metastable minimum, is located at x,. Equation 10 gives the critical stress that a cell can support with a single layer of peptidoglycan. Leduc et al. (1989) observed a thickening of the peptidoglycan layer from 6.6 to 8.8nm during the transition to stationaryphase, as stated previously. It is plausible that a reduction in average cell length is concomitant with this process. If so, it is reasonable to assume that the number of peptidoglycan layers opposing the applied stress is related to the average cell length by
The parameter Lo has been included as a direct inverse proportionality and is unlikely to exist. The total stress resisted by nonseptated cells is simply uu,. This total stress is assumed to be proportional to the mean effective strength, 3,. Introduction of a proportionality constant gives the final form of the correlation for mean effective cell strength:
Regression Results A FORTRAN program using the IMSL routine DRNLIN (modified Levenberg-Marquardt technique) was written to regress the data shown in Figure 2. The regression gave eq 14 as the final correlation. The critical cross-linkage, X L , was found to be 0.568 with a corre-
14.35 - 10.02 In
(")1-x, ( x , I0.568) (14a)
sn= Ln - 237'3 + 2.763
(x,
> 0.568)
(14b)
Discussion and Conclusion Figure 3 showsthat eq 14gives a good unbiased estimate of mean effective strength. Numerically, there appears to be little improvement over eq 5. However, eq 14 is based on a modeling approach to the problem, whereas eq 5 is entirely empirical. Specifically, eq 14 is based on an understanding of how the key parameters (x, andLn) affect strength. Conversely, the relationship between the variables in eq 5 has no basis. It is therefore possible that eq 14 has a greater range of applicability than eq 5. Figure 4 presents plots of the system free energy (eq 6) against cross-linkage for various stress levels, u. The features previously described are clearly visible. We see that a cell will be unstable regardless of its degree of cross-linkage for stress levels greater than a.Below a, we see a shift in the free energy maximum to higher crosslinkages as the stress is raised. Cells with a higher peptidoglycan cross-linkage (less than XL) will therefore be stronger. Equation 14b predicts that the cell obtains no further mechanical benefit in increasing its degree of cross-linkage beyond 56.8% (Sn becomes independent of xc). A further increase beyond x c is unfavorable because of the entropy of mixing term in eq 6. The maximum cross-linkage determined experimentally for this strain is approximately 57.7 % (stationary population), which is within experimental error of XL. To test the significance of each variable grouping in eq 14a, the equation was linearized by taking logarithms. Multiple linear regression (using KT,~a), A, and
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Lo as above) gave In s, = 1.015 In (122.67~~) - 1.08 In (E, + 2.763) +
The maximum standard error estimate in each of the three regression coefficients is approximately 0.1, giving a minimum t-statistic of approximately 10. This is a highly significant result. In conclusion, we have developed a correlation for mean effective strength based on an understanding of cell-wall structure. This correlation gives a good unbiased estimate of mean effective strength. While the final correlation offers no numerical advantage over the linear relationship (eq 5), the variable groupings can be justified.
Notation proportionality constant, eq 13 disruption bond dissociation energy, eq 6 exponent in eq 4 fraction of events which result in a disruptivestress greater than S frequency distribution of effective cell strengths free energy average length of nonseptated bacteria (pm) length parameter, eq 11 constant in eq 4,(MPa-”) exponent in eq 4 homogenizer pressure (MPa) effective cell strength mean effective strength of nonseptated bacteria mean effective strength of septated bacteria temperature fraction of intact bonds limiting cross-linkage septated fraction of the bacterial population
Greek Symbols K bond elastic modulus, eq 6 U stress critical stress on a single peptidoglycan layer uc UL limiting stress standard deviation of the nonseptated bacteria’s an strength distribution standard deviation of the septated bacteria’s 08 strength distribution V peptidoglycan layers opposing the applied stress
Literature Cited Blumberg Selinger, R. L.; Wang, Z.-G.; Gelbart, W. M.; BenShaul, A. Statistical-thermodynamicapproach to fracture. Phys. Rev. A 1991,43(8),4396-4400. Glauner, B.; Hdtje, J.-V.; Schwarz, U. The composition of the murein of Escherichia coli. J.Biol. Chem. 1988,263,1008810095.
Leduc, M.; Frehel,C.; Siegel, E.; van Heijenoort, J. Multilayered distribution of peptidoglycan in the periplasmic space of Escherichia coli. J. Gen. Microbiol. 1989,135,1243-1254. Middelberg,A. P.J.; O’Neill,B. K.; Bogle, I. D. L. A new model for the disruption of Escherichia coli by high-pressure homogenization I. Model development and verification. Trans.Inst. Chem.Eng. Part C (FoodBioprod.Process.)1992a, 70,205-212.
Middelberg, A. P.J.; O’Neill, B. K.; Bogle, I. D. L.; Gully, N. J.; Rogers, A. H.; Thomas, C. J. A new model for the disruption of Escherichia coli by high-pressure homogenization 11. A correlation for the effective cell strength. Trans. Inst. Chem. Eng. Part C (Food Bioprod. Process.) 1992b,70, 213-218. Vervoorn, P. M. M.; Austin, L. G. The analysis of repeated breakage events as an equivalent rate process. Powder Technol. 1990,63,141-147. Verwer, R. W. H.; Nanninga, N.; Keck, W.; Schwarz, U. Arrangement of glycan chains in the sacculus of E. coli. J. Bacteriol. 1978,136,723-729. Accepted November 9,1992.
