Summing up dynamics: modelling biological
processes in variable temperature scenarios
L.M.M. Tijskens *, F. Verdenius
ATO, PO Box 17, 6706 AA Wageningen, The Netherlands
Received 15 December 1999; received in revised form 31 March 2000; accepted 10 May 2000
Abstract
The interest of modelling biological processes with dynamically changing external condi-tions (temperature, relative humidity, gas condicondi-tions) increases. Several modelling approaches are currently available. Among them are approaches like modelling under standard condi-tions, temperature sum models and dynamic modelling. While the ®rst two approaches require huge simpli®cations that endanger the applicability of the results, the latter requires a substantial modelling and computational eort. In this paper the often very successful method of temperature sum is improved and enhanced to re¯ect fundamental insights in biochemical processes. Knowing that reaction rates depend on temperature according to Arrhenius' law, a rate sum calculation for each active process is proposed. While the temperature sum approach is in practice restricted to polynomial models, the rate sum approach allows the building and application of more fundamental and process-oriented models. The method is computation-ally feasible. Model calculations on simulated data show that this approach is at least equivalent to existing approaches, and often outperforms them in terms of statistical ®t (R2
adj
of over 90%, and often 99.5%). Moreover, it has the major advantage of estimating param-eters that have an interpretation in the biochemical reality. Another major advantage is that all the normal rules, techniques and procedures of statistics remain applicable. # 2000 Elsevier Science Ltd. All rights reserved.
Keywords:Dynamic temperature scenarios; Temperature sum; Rate sum; Dynamic modelling; Funda-mental model formulation; Non-linear regression analysis
0308-521X/00/$ - see front matter#2000 Elsevier Science Ltd. All rights reserved. P I I : S 0 3 0 8 - 5 2 1 X ( 0 0 ) 0 0 0 2 7 - 5
www.elsevier.com/locate/agsy
1. Introduction
Developing models for use in agriculture frequently involves working with dyna-mically changing external conditions. This mainly concerns temperature that varies during the growing of produce, as well as during the subsequent storage and hand-ling in the entire food chain. Other external factors that are often dynamic are rela-tive humidity and (solar) radiation. In scienti®c studies, the experimental conditions are kept as constant as possible to avoid the troublesome dynamics in interpreting and modelling the results. Especially in agricultural research, control of external factors like temperature, relative humidity and radiation intensity is often impos-sible. This is not only the case during production in open ®eld, but also in many parts of distribution chains. Consequently, these external factors vary too much to consider them as constant. The dynamics of the environment have to be approxi-mated to include their eects in models and data analysis. Existing approaches for these approximations either make radical simpli®cations, such as assuming mean
Nomenclature
Variables
C Concentration of compound or property
Ea Energy of activation
Enz Enzyme activity
k Rate constant
kS Rate integral or rate sum
P Concentration of some product of reaction
R Universal gas constant
R2
adj Percentage variance accounted for
T Temperature
t Time
TS Temperature sum
Indices
0 Initial value att=0
1,2,3 Occurrences of concentrations or processes
d Denaturation
i,j Running array indicators
n Number of observations
na Not active
p Process number indicator
oset Applied oset
ref At reference temperature
values for external conditions, or result in hard-to-interpret results, as obtained in polynomial or purely statistical modelling.
In this paper a new and fundamental approach is developed and described to deal with dynamic temperature scenarios. The eect of temperature on rates of occurring processes as described by Arrhenius' law is included, without harsh simpli®cations or assumptions. The potential of its application is indicated with a few examples.
2. Describing the problem
In modelling the behaviour of products and processes, the primitive, most basic functions usually applied are dierential equations. Often analytical solutions can be derived for these sets of dierential equations for constant external conditions as, for example, temperature. These analytical solutions can be used for statistical ana-lysis of experimental data, obtained at constant conditions. And these are exactly the conditions that applied statistics have promoted for experimental research to be conducted. There is a large set of robust tools and analysis schemes from applied statistics to facilitate this approach.
On many occasions, however, it is impossible to conduct experiments at constant external conditions. Only in well-controlled production and storage facilities, e.g. greenhouses, does the assumption of constant conditions hold. In outdoor farming, however, control of temperature, rainfall or solar radiation is no option. Hence, in practice, conditions are highly dynamic. Applicability of the derived model, func-tions and results is therefore very limited. To make at least an attempt to use these valuable data several approaches are applied, all having some drawbacks on relia-bility or practical applicarelia-bility.
3. Traditional approaches
3.1. Applying mean temperature
When the variation in temperature is not all too large, a possible and often-applied approach is to use the mean temperature during the period of experiment-ing, as a substitute for the real occurring temperature. In storage experiments, for example, the temperature in controlled storage rooms does ¯uctuate around the set point. The ¯uctuations are usually small with an order of magnitude of one to two degrees. Also, the eect of these temperature ¯uctuations is normally considered less important then the eect of product variance and measuring error. So, the tem-perature ¯uctuations are almost never taken into consideration in the statistical analysis of the experimental results, and the classical analysis for constant condi-tions is applied. The reliability of the results is suciently high for a sensible and useful interpretation.
dC t
dt ÿk T t C t : 1
At constant temperatures,T(t) is constant and the dynamic rate constant [nota-tion k(T(t))] becomes independent from temperature and consequently from time (notationk). Upon integration at constant conditions, the well-known exponential function emerges:
C t C0eÿkt: 2
The advantage of this approach is its ease of operation, and the reliable techniques available for statistical analysis. The disadvantage is, of course, the fact that eects of temperature ¯uctuations are cancelled out before their insigni®cance can be proven.
3.2. Applying dynamic modelling
When the dierences in temperature are too large to neglect, the most sensible approach is to use for statistical analysis directly the dierential equations and the eect of temperature on the parameters of the dierential equations. In this case, Eq. (1) would be used by combining a numerical integration with a statistical esti-mation procedure. The mathematics of the applied technique looks like:
t
t0
dC t ÿ t
t0
k T t C t dt: 3
This is the technique used in simulation programs and modelling languages. The advantages of the technique are manifold. In the ®rst place, no further assumptions are made to arrive at a statistical estimation of the value of the parameters, except for the numerical integration. Good and reliable numerical integrators are amply available nowadays. In the second place, there is no need to obtain an analytical solution, which is often dicult or even impossible. The derived dierential equa-tions are used as is. In the third place, the sampling or measuring frequency does not need to be at regular intervals. In the last place, dynamic experiments can provide a very neat way to decrease the number of experiments to be conducted, and still obtain the same amount of information.
There are, however, also drawbacks to the technique. Combinations of numerical integrators with statistical robust optimisation algorithms are not readily available. Most of the time a dedicated program has to be written (and tested) based on the actual model and the actual data to be used. Examples of this technique can be found in Verlinden et al. (1995, 1996, 1997). Some modelling languages are emerging that are capable of performing parameter estimation on dierential equations, e.g. Modelmaker (Cherwell Scienti®c Ltd, Oxford, UK). However, these systems are not yet capable of handling complex and heavily interactive models.
dicult to make clear where and how much information on temperature can be obtained from a particular scenario. Designing scenarios to obtain the most ecient way for conducting dynamic experiments is rapidly becoming a new subject of sci-enti®c study (Versyck et al., 1997a, b, 1998a, b, 1999; Bernaerts et al., 1998, 2000; Van Impe, 1998; Van Impe et al., 1998).
3.3. Applying temperature sum
Another frequently used technique in black box or empirical modelling is the application of the temperature sum over time. In calculating this temperature sum, an oset is sometimes considered to cut o temperatures that are known to be without eect. For a series of collected temperaturesTiat a series of distinct timesti,
the temperature sum (TS) is at each time calculated as the sum of all temperatures up to the actual time. This can mathematically be represented as:
TSiX
This temperature sum is frequently used in modelling the observed behaviour of some variable with second- or even higher-order polynomials or to apply combined time± temperature information in existing models (De Visser, 1992; D' Antuono and Ros-sini, 1994; Grevsen, 1998). The temperature sum has been applied in combination with machine learning approaches in an application to manage product inherent variance (Verdenius, 1996). The temperature sum technique has proven to be reliable and generically applicable in the pre- and postharvest chain of agricultural products. Some reason has to exist why this empirical approach is so general applicable.
By applying this kind of transformation a new pseudo-variable (TS) is created that contains all the information on all combinations of time and temperature occurring in the experiments. The observed behaviour of the phenomena under study is now explained based on this pseudo-variable applying mostly higher-order polynomials. Of course, relations with processes that are more fundamental and fundamental laws that could be applicable are lost as one does absolutely not know how this pseudo-variable could possibly govern the occurring processes. The parameters of such empirical models lack all physical, chemical, biochemical or physiological meaning. Results are therefore hard to interpret, and impossible to translate to circumstances dierent from the experimental ones.
4. Alternative approach
At a closer inspection of Eq. (4), one can readily see that the temperature sum is a crude approximation of the integral of the variable temperature over time:
TS t t
t0
T t ÿToffset
The relation depicted in Eq. (5) also makes completely clear the importance of suf-®cient sampling frequency, to make sure that the information contained in the measured temperature data is sucient to approximate, by integration or summa-tion, the real integral of temperature over time: is the applied temperature scenario suciently described by the collected temperature data.
On closer comparison of Eqs. (3) and (5), it becomes evident where the drawbacks of the application of temperature sum originate. In the theoretical relation (Eq. (3)), the eect of temperature on the rate of processes is included before the integration, whereas in the practical relation (Eq. (5)) that eect will be applied after integration (or summation) of temperature over time.
As the processes occurring in living or even in dead biological produce are in most cases chemical and biochemical processes, the rates of reaction will depend on tem-perature, presumably according to the law of Arrhenius:
kpkp;refe
wherepindicates the process involved. With dynamic temperature scenarios applied to the biological material, this equation extends to:
kp T t kp;refe
Instead of integration over temperature directly, one should integrate for each occurring process separately over the rate constant, as deduced in Eq. (3). This results in a pseudo-variable kS, speci®c for each occurring process:
kSp t t
0
kp T t dt: 8
When the dierential equation for a ®rst-order exponential process (Eq. (1)) is integrated with variable temperature included, one obtains (Pinheiro Torres and Oliveira, 1999):
C t C0eÿ
t
0kp T t dt: 9
The integral in the exponent is the same as the rate sum (Eq. (8)). For practical application of this deduction, in the analytical solution (Eq. (2)) at constant external conditions (of temperature), the term kpt can be replaced by the so-obtained
kSp , which in turn can be approximated by a summation:t
The term kp T1constitutes a correction factor, similar to the trapezoidal rule for
approximating an integral with a summation. This correction factor ensures that kS at the ®rst measuring point is zero.
The same fundamental approach to include the eect of temperature history in model formulations based on application of Arrhenius' law was used by Wells and Singh (1988). They, however, developed the idea further to estimate an eective tem-perature as a replacement for the mean temtem-perature. In addition, Allen (1988) already strongly reported that the mean of a function is de®nitely dierent from the function of the mean. In our case, the function is Arrhenius. Therefore, the rate at mean tem-perature is dierent from the mean of the rates over the dierent temtem-peratures.
So, instead of using a general temperature sum in empirical modelling, one can apply much more fundamental models using a rate sum speci®c for each process. These fundamental models have a bearing on the processes occurring in the product. Using these more fundamental models instead of empirical models, the under-standing of what is going on in our products, the development of theoretical views, the development of tools and systems increases very much. The scope of practical application of models and theories is greatly enlarged (Tijskens et al., 1998a).
5. Theoretical examples
5.1. Materials, methods and data generation
All data used in this paper are simulated data based on models of simple reaction mechanisms by numerical integration of the dierential equations. All mathematical developments of models and generation of simulation data were conducted with MapleV release 4 (Waterloo Maple Inc., Waterloo, Canada). Statistical analysis was conducted with the non-linear regression procedure of Genstat 5 (Rothamsted, UK). For the next examples, data were generated by simulation using imaginary but plausible values for the model parameters on three basic and simple models, which occur in nature and are frequently applied in modelling of processes in agricultural products. For each of the models, three temperature scenarios were applied, a simple heating up and cooling down over 20C, a sinusoidal change in temperature, to mimic daily temperature ¯uctuations in the growing ®eld, and a simple scenario to mimic the temperature in a food distribution chain. In Fig. 1 the three tempera-ture scenarios are shown.
5.2. Simple ®rst-order degradation
Simple ®rst-order degradation is a process very frequently occurring in nature, and applied commonly in modelling. The mathematical equations are already pro-vided (Eqs. (1)±(9)) and used in the mathematical deduction. The mechanism is a simple conversion:
For this process, the system of rate sum is fundamentally correct (Eqs. (8)±(10)) and does not introduce another error except the approximation of an integral by a summation.
In the analytical solution at constant temperature (Eq. (2)) the term ktmay be substituted by the rate sum as de®ned by Eqs. (8) and (10). This results in:
C t C0eÿkS t: 12
This signi®es that this system can and may be applied at any time, provided the information contained in the data, covering both the behaviour of the state variable
C, and the eect of temperature on the rate of reaction, is sucient to detect all these eects. But that is a golden rule in any statistical analysis: if the data do not contain information on a certain subject, no conclusions can be drawn from these data on that subject.
Data were generated by numerical integration of Eq. (1), using the three dynamic scenarios, and a value for the input parameters as shown in second column in Table 1. By applying the approximation of the rate sum kS(t) for each of the dynamic scenarios instead of the factorkstthe dynamics of the temperature scenarios can
be analysed using the standard statistical routine of non-linear regression analysis.
Table 1
Results of statistical data analysis with rate sums to incorporate dynamic scenariosa
Input Heat-cool Daily Distribution All scenarios combined
Estimate S.E. Estimate S.E. Estimate S.E. Estimate S.E.
R2
adj 100 100 99.9 100
C0 100 100.132 0.0791 100.925 0.117 98.6734 0.0555 99.617 0.0784
ks,ref 0.06 0.060402 0.000288 0.060667 0.000172 0.055316 0.000428 0.059385 0.000104
Eas 10 0000 10 197.7 79 9758.1 84.2 9671 45.7 9951.5 24
a Model experiments on ®rst-order reactions with exponential decay. S.E., standard error of estimate. See Nomenclature for de®nition of other abbreviations.
In Table 1 the statistical estimates of the input parameters are shown for each dynamic scenario. For this simple exponential behaviour, the rate sum system is fundamentally correct. The only approximation applied is found in the summation of the rate, instead of integration. The results are therefore not surprisingly com-pletely correct, as can be taken from the estimated values for the parameters, their standard errors and the extremely high percentage variance accounted for (R2
adj).
The three temperature scenarios were applied to the exponential model with the same input values for the parameters (second column in Table 1). This means that the data can be pooled together and analysed in their entirety, again using the standard statistical procedures of non-linear regression. This technique of pooling data takes advantage of the larger set of data, and the better description of the temperature dependency in the three scenarios together. The results of the non-linear regression analysis are shown in the last two columns of Table 1.
The ®rst-order exponential decay is also a part of the next examples on the activity of a denaturating enzyme and on the consecutive reaction. The estimated value of the parameters of the exponential enzyme denaturation analysed with non-linear regression analysis using the exponential formulation (Eq. (12)) can be taken from Table 2. The estimated values of the parameters of the exponential conversion of C1 into C2 analysed with the same technique are shown in Table 5. For each of the three applied scenarios, the parameter used can be estimated very well. The tem-perature information in the distribution scenario is, however, too low to obtain a really reliable estimate. This scenario contains only information in the region around 20C and around 2C (Fig. 1). It is obvious that two data points are too few for an allowed and reliable regression analysis.
5.3. Enzymatic conversion
The second example is the action exerted by a denaturating enzyme. This type of behaviour frequently occurs when applying moderate heat treatment (e.g. blanching)
Table 2
Results of statistical data analysis with rate sums to incorporate dynamic scenariosa
Input Heat-cool Daily Distribution
Estimate S.E. Estimate S.E. Estimate S.E.
R2
adj 99.9 99.9 99.5
Enz0 1 0.999974 1.34E-05 1.004558 0.000983 0.9978 NA
kd,ref 0.001 0.001007 1.29E-05 0.010228 0.000214 0.009142 NA
Ead 25 000 25 381 387 24 575 433 26 011 NA
R2
adj 100 100 100
S0 100 99.9344 0.0545 99.904 0.138 98.596 0.0911
ks,ref 0.1 0.100438 0.000177 0.09507 0.000217 0.09916 0.00103
Eas 5000 5135.7 19.6 6360.5 70.9 4968.8 46.2
to agricultural products (Tijskens et al., 1999). The reaction mechanism is basically a combination of a ®rst-order reaction and a second-order reaction as shown in Eq. (13):
Enz!kd Enzna;
SEnz!ks PEnz: 13
A set of dierential equations can be derived based on the fundamental laws of reaction kinetics. This set of dierential equations is used to generate the simulated data (Eq. (14)).
@Enz t
@t ÿkdEnz t
@S t
@t ÿksEnz t S t
@P t
@t ksEnz t S t 14
The analytical solution of this set of dierential equations at constant temperatures is:
Enz t Enz0e ÿkdt;
S t S0e
Enz0 ks
kd e
ÿkdt
ÿ1
ÿ
: 15
Data were generated by numerical integration of Eq. (14), using the three dynamic scenarios, and a value for the input parameters as shown in second column in Table 2. The behaviour of the substrateSat the three dynamic scenarios is shown in Fig. 2.
The factorskstandkdtcan then again be approximated by the rate sums kSs
and kSdto include the dynamics of the temperature scenarios in the analysis. That
leaves the unresolved meaning of the rate constants ks and kd that occur in the
equation without being coupled to the time t(ks/kd). These parameters apparently
have a dierent meaning in the rate sum approximation of the temperature dynam-ics than in the analytical solution and should re¯ect the dynamdynam-ics of temperature. As these rate `constants' in analytical solutions always occur in ratios, the time is can-celled out and they can be replaced by their respective rate sums: (kSs/kSd).
In Table 2 the statistical estimates of the input parameters are shown for each dynamic scenario. In the ®rst analysis (top half of the table), the parameters for the enzyme denaturation were estimated based on information contained in the gener-ated data of Enz(t) only. In the second analysis (bottom half of the table), the parameters of the enzymatic action were estimated, keeping the value of the para-meters of the denaturation process constant as estimated in previous analysis. The parameters for both model elements were estimated for all three scenarios reli-ably: the percentage variance accounted for (R2
adj) was 99.5 up to 100% (Table 2).
Estimation of all parameters simultaneously only on values for S(t), assuming no information whatsoever about parameter values of the denaturation process, proved dicult. That is more a sign that the more measured data are available the better the estimation procedure can work, rather than that this approach to dynamic beha-viour is not working properly.
The data of this theoretical example are also analysed using the temperature sum in polynomial models, up to a fourth-degree polynomial. In Table 3, the results are shown of the analysis of the data on substrate during application of the daily sce-nario, analysed with increasing order of polynomial. The percentage variance accounted for (R2
adj) is acceptable from the third order upward. However, the values
of the coecients are clearly dierent between successive orders of the polynomial. As a consequence, the order of the polynomial chosen determines not only the reliability but the applicability as well.
In Table 4, the results are shown for the classical analysis of substrate as a function of a fourth-order polynomial in temperature sum. Although each of the scenarios is
Table 3
Results of statistical data analysis with increasing order of polynomial in temperature sums to on one dynamic scenarioa
described reliably by the model, the parameter values for each of the separate sce-narios are very dierent, sometimes up to one order of magnitude. As a con-sequence, the results of the analysis of one scenario cannot be used to described or predict the behaviour of the substrate in another scenario. This limits the generic applicability of these types of models.
5.4. Consecutive reactions
Consecutive reactions do also occur in nature and are used in fundamental-oriented models (Tijskens et al., 1998b). The simpli®ed mechanism is basically a combination of two ®rst-order reactions as shown in Eq. (16):
C1!
k1 C2;
C2!
k2
C3: 16
Based on this mechanism and applying the rules of fundamental kinetics, one arrives at a set of dierential equations:
@C1 t
@t ÿk1C1 t;
@C2 t
@t k1C1 ÿt k2C2 t: 17
The analytical solution at constant temperatures is:
C1 t C10eÿk1t;
C2 t C20eÿk2tC10k1
eÿk1tÿeÿk2t
k2ÿk1
: 18
Table 4
Results of statistical data analysis with fourth order of polynomial in temperature sums on all three scenariosa
Model scenario Substrate
Heat-cool estimate Daily estimate Distribution estimate
Nobs 80 80 80
R2
adj 99.8 100 92.5
Constant 95 99.702 120.46
TS ÿ0.3354 ÿ0.2347 ÿ1.1696
TS2 7.5593E-4 2.7371E-4 40.462E-4
TS3 ÿ9.4410E-7 ÿ1.6107E-7 ÿ18.045E-7
TS4 4.5093E-10 0.36921E-10 ÿ71.018E-10
Again, data were generated by numerical integration of Eq. (17), using the three dynamic scenarios, and a value for the input parameters as shown in second column in Table 5. The behaviour of the intermediate compound C2 at the three dynamic
scenarios is shown in Fig. 3. Again, the factorsk1tandk2tcan be approximated
by their respective rate sums kS1(t) and kS2(t). The unresolved occurrences ofk1and
k2, not coupled with time, only occur in ratios. They can therefore again be
sub-stituted by the respective rate sums. The dynamics of the temperature scenarios can now be analysed by non-linear regression. In Table 5 the statistical estimates of the input parameters are shown for each dynamic scenario for each equation of Eq. (18), analysing both reactions simultaneously and the ®rst reaction separately.
Table 5
Results of statistical data analysis with rate sums to incorporate dynamic scenariosa
Input Heat-cool Daily Distribution All scenarios combined
Estimate S.E. Estimate S.E. Estimate S.E. Estimate S.E.
R2
adj 100 93.4 99.9 99.7
Nobs 81 81 81 243
C10 80 79.9927 0.00763 79.9722 0.0104 79.876 0.486 79.87 0.311
k1ref 0.5 0.486461 0.000325 0.472767 0.000149 0.59644 0.00951 0.52911 0.00628 Ea1 10 000 10 027.57 4.64 10 359.6 15.2 10 698.5 81.1 10 094.8 72.9
R2
adj 100 100 100 99.9
Nobs 81 81 81 243
C10 80 80.407 0.198 97.337 0.22 Many 83.314 0.681
C20 20 19.9099 0.0493 3.705 0.124 19.383 0.302
k1ref 0.5 0.45343 0.00714 0.44875 0.002 0.5247 0.00909
Ea1 10 000 9695.6 74.3 8752 273 10 132.9 75.1
k2ref 0.15 0.16508 0.00117 0.14684 0.00116 0.15454 0.00137
Ea2 15 000 15 856.9 87.5 16 900 253 14 539 85.4
a Model experiments on consecutive reactions. S.E., standard error of estimate. See Nomenclature for
de®nition of other abbreviations.
Again, the estimates for all three scenarios are very satisfactory. For the distribution scenario, multiple solutions are possible, at least for the statistical estimation pack-age. This con®rms once again the fact that insucient temperature information is contained in the data of this scenario: roughly only two temperature levels are present in the scenario. These ®ndings again stress the importance of scenario design in estimating parameters in a dynamic environment (Versyck et al., 1997a, b, 1998a, b, 1999; Bernaerts et al., 1998, 2000; Van Impe, 1998; Van Impe et al., 1998).
Since the processes, occurring in the three dierent scenarios, are fundamentally the same. It should therefore be possible to analyse the data of these three scenarios together. The rates sums have then to be calculated over the temperatures of each scenario separately. In the last column of Table 5 the results of the combined ana-lysis are shown. Now the amount of information on temperature±time combinations is amply sucient to obtain reliable and unique estimates of the parameters of the model. This exercise proves the added value of the principle: models based on fun-damental relations are indeed reusable in all kind of scenarios.
6. Conclusions
The empirical temperature sum in polynomial modelling can be replaced by a rate sum, speci®c of each occurring process. The approach proposed allows the devel-opment of more fundamental models that can still be applied in dynamic external factors.
Statistical analysis of the data incorporating all information on the dynamic sce-narios is possible on a function level, and no dierential equations need to be used. This ensures that the huge package of statistical tools remains available for appli-cation in dynamic external conditions.
In dynamic circumstances, multiple solutions for parameter estimation are pos-sible. This makes the understanding of interactions and of eects even more impor-tant, and a larger emphasis has to be put on the experimental design.
Acknowledgements
This study was partially conducted in the framework of the SPOT-IT2 project. The ®nancial support of FTK Holland BV, Bleiswijk, The Netherlands, is gratefully acknowledged as are the valuable comments and suggestions of Prof. Dr. J. Van Impe, Dr. J. Top and Dr. J.M. Soethoudt.
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