- Email: [email protected]
Flocculation in brewing yeasts: A computer simulation study
BioSystems 83 (2006) 51–55
Flocculation in brewing yeasts: A computer simulation study M. Ginovart a,∗ , D. L´opez b , A. Gir´o b , M. Silbert c,d a b
Departament de Matem`atica Aplicada III, Edifici ESAB, Universitat Polit`ecnica de Catalunya, Campus del Baix Llobregat, Avda. Canal Ol´ımpic, 08860-Castelldefels, Barcelona, Spain Departament de F´ısica i Enginyeria Nuclear, Edifici ESAB, Universitat Polit`ecnica de Catalunya, Campus del Baix Llobregat, Avda. Canal Ol´ımpic, 08860-Castelldefels, Barcelona, Spain c Departamento de F´ısica, Universidad de Extremadura, 06071 Badajoz, Spain d Institute of Food Research, Norwich Research Park, Colney, Norwich NR4 7UA, UK Received 27 July 2005; received in revised form 14 August 2005; accepted 5 September 2005
Abstract A new simulator, INDISIM-FLOC, based on the individual-based simulator INDISIM, is used to examine the predictions of two different models of yeast flocculation. The first, proposed by Calleja [Calleja, G.B., 1987. Cell aggregation. In: Rose, A.H., Harrison, J.S. (Eds.), The Yeasts, vol. 2. second ed., Academic Press, London, pp. 165–238 (Chapter 7)], is known as the “addition” model. The second, proposed by Stratford [Stratford, M., 1992. Yeast flocculation: a new perspective. In: Rose, A.H. (Ed.), Advances in Microbial Physiology, vol. 33. Academic Press, London, pp. 1–71] is known as the “cascade” model. The simulations show that the latter exhibits a better qualitative agreement with available experimental data. © 2005 Elsevier Ireland Ltd. All rights reserved. Keywords: Discrete simulation; Yeast flocculation; Individual-based model; Addition model; Cascade model
1. Introduction Yeast flocculation is the phenomenon of cellular aggregation when cells adhere, reversibly, to one another to form macroscopic flocs. It is an important process for the production of beer that causes the yeast to sediment, or cream, at the end of the fermentation process. Thus, the yeast can be harvested from the bottom (lager fermentation) or top (ale fermentation) of the fermenter and used for the next fermentation. Yeast flocculation remains an active field of research (for a recent, short, review see Verstrepen et al., 2003). Whilst there appear to be similarities with the process of sedimentation, or creaming, in colloidal disper-
∗
Corresponding author. Tel.: +34 93 5521133; fax: +34 93 5521001. E-mail address: [email protected] (M. Ginovart).
sions, the mechanisms involved are different (Stratford, 1992). In fact, yeast flocculation is a very complex process that depends on the expression of specific flocculation genes (Bony et al., 1998). Turbidimetric tests suggest that flocculation in bottom fermenting, but not necessarily in top fermenting, strains is mediated by a lectin-aggregating mechanism (Dengis et al., 1995; Ngondi-Ekome et al., 2003). Actually, yeast flocculation is a process in which biological, chemical and physical processes are involved. In this work, we are concerned mainly with the latter. In general, the flocculation of brewing yeasts is associated with the onset of the stationary phase, but the timing and degree of flocculation is not regulated precisely. This is an important consideration in brewing practice because if flocculation occurs too early, fermentation will cease prematurely, leaving residual sugar in the wort. This can cause microbiological stability problems and
0303-2647/$ – see front matter © 2005 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2005.09.002
52
M. Ginovart et al. / BioSystems 83 (2006) 51–55
can adversely affect flavour characteristics of the final product. Conversely, if it takes an unduly long time for cells to flocculate, this can cause downstream processing problems in beer clarification; see, e.g., Stratford (1992); Walker (1998). Flocculation may take place as a result of natural (in unstirred vessel) or external (mechanical) agitation. In this work, we are only concerned with the latter. It has been shown experimentally that flocculation requires an agitation threshold, namely a minimum kinetic energy to overcome the electrostatic, steric or hydrophobic repulsion between yeast cells or cell aggregates (Stratford and Keenan, 1987). After flocculation is started, the process advances rapidly to a new stationary state, namely a dynamical equilibrium between the rate of formation of new flocs and the rate of floc dissociation. It has also been observed that the size distribution of the flocs depends on the intensity of agitation. On the one hand the intensity of agitation favours the formation and growth of the flocs, while on the other it also contributes to their rupture through shearing (Stratford et al., 1988). Basically, in this steady state the system contains two fractions: flocs and single cells. We are concerned here with the simulation of the flocculation processes of a model yeast Saccharomyces cerevisiae using a new individual-based computer simulation code, INDISIM-FLOC, based on the generic simulator INDISIM1 (Ginovart et al., 2002), which we have developed to study bacterial growth. We are interested specifically with the consequences of the frequency of floc–floc collisions in the association/dissociation rates of flocs. As a result, the simulation of flocculation requires important changes to the computer code used in our previous yeast modelling studies (Ginovart et al., submitted for publication). The simulation of yeast growth and flocculation involve two different length and time scales, compared to those used in our previous studies of yeast growth, which have to be treated separately. Hence, whereas in the simulation of yeast growth the code is capable of controlling the characteristics of each cell, in the simulation of flocs it is only possible to control the number of “cells” in each floc. This, in turn, demands a new computer code, which we call INDISIM-FLOC, which will be described in Section 2. The changes in scale require that one single unit is made up of about O (102 ) individual yeast cells, which we henceforth call an “individual cell”, and that the unit of time is much larger than colli1
INDISIM is not publicly available at present. Please contact the corresponding author for specific information about our simulator and its applications.
Fig. 1. Experimental data (after Brohan and McLoughlin, 1984). Size distribution of the flocs as a function of their magnitude of the brewing yeast S. cerevisiae: () 250 rpm; (䊉) 500 rpm; () 750 rpm; () 1000 rpm.
sion times. These changes in scaling, when dealing with physical systems, are known as coarse graining, a term that we use here as well. Several recent theoretical models have been put forward to study different aspects of yeast flocculation by means of continuous models; see, e.g., van Hamersveld et al. (1998); Hsu et al. (2001). In this work, we examine, by means of computer simulations, two earlier models of yeast flocculation. The first, proposed by Calleja (1987), suggests that clustering is an additive process (addition model); the other, put forward by Stratford (1992), suggests a cascade process yielding fractal structures. We compare the results of our simulations, for each of the proposed mechanisms with the experimental results for one strain of the yeast S. cerevisiae (CIIIF) reported by Brohan and McLoughlin (1984), shown in Fig. 1; specifically the distribution of floc sizes as a function of the intensity of the agitation, and also with the experimental results reported by Stratford et al. (1988), for the strain S646-1B, specifically their Fig. 4 (not shown here) describing the effect of agitation on the number of cells within flocs. We discuss the experimental data in conjunction with our simulation results in Section 3. In fact, we suggest that our simulations may be regarded as com-
M. Ginovart et al. / BioSystems 83 (2006) 51–55
puter “experiments” with which we examine – within the set of rules used in the simulations – the predictions of the two models vis a` vis experiment. Our simulations are carried for a system of “elements” that includes both free individual cells and flocs. We define a floc as an aggregate of individual cells; our simulator controls each constitutive individual cell. The elements are placed in a cubic box which, in turn, is divided into small cubic spatial cells. The elements are allowed to move both within and between the small cubic cells. The simulator is set up so that the elements satisfy given rules of motion and of interaction. In fact, the simulator includes rules for the external agitation, the random motion of the elements and their collision. The following two comments are in order. First, within this approach, differentiation between different strains of the same yeast or between different types of yeast is limited to size and shape of the individual cells, as well as the strength and functional form of the aggregation and/or dissociation probabilities. Hence our conclusions regarding the two proposed mechanisms are generic rather than specific. Second, it is assumed that the simulated yeast is flocculent. In the next section, we report briefly on the main new features of the simulator used for this study. Then we present, and briefly discuss, some of the results of our simulations. 2. The simulator We assume an initial population of individual cells towards when the flocculation process starts, and such that: (1) The external (mechanical) agitation A to which the system is subjected is characterized by a parameter that we assume to be constant throughout the simulation. This parameter may be assumed to be the necessary kinetic energy contribution to overcome the repulsive interactions between the flocs and conditions the behaviour of the system. Whence A ∝ kinetic energy ∝ w2 . A is fixed at the beginning of the simulations and it will determine the probabilities of aggregation or dissociation of flocs. (2) The simulator controls flocs of various sizes containing n cells, including individual cells (n = 1). Henceforth the term “floc” also includes the case of an individual cell. Flocs are assumed to exhibit spherical symmetry, and are made up with n, closely packed, individual cells. The starting point for the simulations is an initial configuration of single cells randomly distributed in the spatial domain.
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(3) The spatial domain is a cube subdivided into cubic cells, subject to periodic boundary conditions. Each spatial cell may contain one, several or no flocs. Our simulator controls each constitutive individual cell. Time is divided into equal intervals associated to computer steps. (4) At each time step, the simulator controls the size (n) of each floc and its position in space, characterized by a vector. Each floc is displaced to a new position randomly chosen within a given range R. We assume for this work that this range has a constant value, R = k. There are other possibilities that will be explored in another work, namely (i) that the range is a linear function of the rate of agitation, R = k A or (ii) that the range depends not only on A but also on the mass of the floc involved, say R = k (A/m)1/2 . We intend to probe into these possibilities and will report our simulation results on completion. (5) When a floc moves into a new small cubic cell, the simulator checks whether the spatial cell is occupied with other flocs. If so, one is chosen randomly; the simulator assumes that the two have collided and proceeds to estimate whether the two flocs aggregate or dissociate. (6) In order to compare between the two flocculation models (addition and cascade) we assume in the case of the cascade model, but not in the case of the addition model, that the colliding flocs are of similar size. (7) The following two probabilities are evaluated: (a) aggregation probability Pa and (b) dissociation probability Pd . In both addition and cascade models, the simulator assumes Pa to be constant, and the dissociation probability Pd is related to the exponential of A (the exponential of kinetic energy), and to the size n of the floc. If there is dissociation the distribution of sizes of the two resulting flocs are determined randomly. (8) At each “collision” the simulator evaluates sequentially: (a) Pa ; (b) if there is aggregation, the simulator evaluates the Pd of the resulting floc; (c) if there is no aggregation, the simulator evaluates separately Pd for the two “colliding” flocs. (9) As implied in (5), the constituents of the system do not remain static in their spatial cells. Here, we assume that their motion is solely due to the external agitation. 3. Results and discussion We are concerned here with the relationship between the intensity of agitation and the size distribution of flocs
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M. Ginovart et al. / BioSystems 83 (2006) 51–55
Fig. 2. Simulation results showing the effects of the agitation on the size of the flocs: (a) cascade model and (b) addition model.
and, with our simulator, we have examined two different mechanisms: the addition and the cascade models. We start with Fig. 2, which shows the simulation results for the number of individual cells per floc as a function of agitation. We find that this number diminishes with increasing agitation, in agreement with experiment (Fig. 4 in Stratford et al., 1988). Between 110 and 140 rpm, the experimental data exhibit an abrupt fall in floc size. This sharp fall appears in the simulations for the cascade model (Fig. 2a), but not in the addition model where floc size decreases continuously with agitation (Fig. 2b). There is, however, an important difference in the abrupt fall of floc size between experiment and simulation of the cascade model. In the latter, we find two abrupt falls; one at A1/2 ∼ 1.5 (corresponding to, approximately 144 rpm) and an even sharper fall at A1/2 ∼ 3 (≈617 rpm). Whichever way, the comparison between experimental data and simulations appears to favour the cascade model. We now turn to Fig. 1 which shows the experimental data of Brohan and McLoughlin (1984) displaying the effect of agitation on one of the distributions of floc size of the yeast S. cerevisiae (CIIIF). At the lower agitation rates (250 and 500 rpm), there is a wider distribution of floc sizes, with the lower containing a relatively larger percentage of large flocs. At the higher agitation rates (750 and 1000 rpm), the distribution of floc sizes is more limited; the narrower distribution corresponding to 1000 rpm. Note that only the points are experimental data; the lines joining them are guides to the eye. The authors do not include error bars in their plots. However, assuming that similar errors to those shown in Fig. 4 of Stratford et al. (1988) have been made in estimating floc sizes, then we expect these errors to be larger at the
higher agitation rates. Whence the error bars are probably more relevant in those cases when the distribution of floc sizes is narrower. Our simulations are intrinsically exact. Error bars in our simulations may be obtained by repeating the simulations with different initial configurations; this was not done for this work, but are assumed to be very small. We now turn to the simulations, shown in Fig. 3. Since we originally carried out our simulations in terms of A1/2 we used those simulation values, in rpm, nearest to the experimental data. Fig. 3a shows the results of our simulation using the cascade model, namely when only flocs of similar sizes are allowed to collide; while Fig. 3b shows the results of our simulations using the addition model. The cascade model simulations, Fig. 3a, shows
Fig. 3. Size distribution of the flocs as a function of their magnitude of the brewing yeast S. cerevisiae from the simulations: () 310 rpm; (䊉) 481 rpm; () 723 rpm; () 1012 rpm. On the left with the cascade model and on the right with the addition model.
M. Ginovart et al. / BioSystems 83 (2006) 51–55
a wider range of size distributions than the experimental data, at both 250 and 500 rpm. On the other hand, for the same agitation values, the size distribution in the addition model, Fig. 3b, is narrower than the experimental data, even allowing for error bars similar to those indicated in Stratford et al. (1988). Moreover, and also in agreement with experiment, the distribution of larger flocs at 250 rpm in the simulations of the cascade model, is higher than for 500 rpm. Except for one value this is not the case with the addition model. The comparison between Fig. 1 and Fig. 3(a and b) again suggests that the cascade model exhibits a better qualitative agreement with the experimental data. We conclude that our simulations validate the cascade model, within the rules used in the simulations reported above. Finally, the following comment is in order. Our initial population does not consider the distribution of genealogical ages which, in our simulations, may be modelled by a distribution of “scars” on the surface of individual yeast cells (Ginovart et al., submitted for publication). Moreover, there is an intrinsic difficulty in relating the distribution of scars in yeast cells with the assumptions that need to be made in order to say something about the distribution of genealogical ages in individual cells (which contain O (102 ) yeast cells). There is, however, recent evidence that aged cells display an enhanced flocculation potential (Powell et al., 2003). We hope to be able to surmount the difficulties involved to incorporate in our simulator sensible assumptions that allow us to take the individual cell average age into account. There is also the added difficulty as to how we include the extra degree of “stickiness” in individual cells that, on average are deemed as “aged”, and what would be the case when a “young” and an “aged” individual cells collide. We are studying this problem at present and, if we succeed in overcoming these difficulties, will report our results on completion. Acknowledgements We thank J. Valls for valuable discussions and their general interest in this work. One of us (MS) has been supported by a Marie Curie Fellowship of the IHP European Community Programme under Contract no. MCFI-
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2002-00081. This work has been partially supported by DURSI, Generalitat of Catalonia (2003ACES0064) and by Ministry of Education and Science of Spain (DGYCIT PB97-0693; GL2004-01144). We are grateful to the anonymous referee for critical comments and suggestions that, we hope, have resulted in an improved manuscript. References Bony, M., Barre, P., Blondin, B., 1998. Distribution of the flocculation protein, Flop, at the cell surface during yeast growth: the availability of flop determines the flocculation level. Yeast 14, 25– 35. Brohan, B., McLoughlin, A.J., 1984. Characterization of the physical properties of yeast flocs. Appl. Microbiol. Biotechnol. 20, 16– 22. Calleja, G.B., 1987. Cell aggregation. In: Rose, A.H., Harrison, J.S. (Eds.), The Yeasts, vol. 2, second ed. Academic Press, London, pp. 165–238 (Chapter 7). Dengis, P.B., Nelissen, L.R., Rouxhet, P.G., 1995. Mechanisms of yeast flocculation comparison of top fermenting and bottom fermenting strains. Appl. Environ. Microbiol. 61, 718–728. Ginovart, M., L´opez, D., Valls, J., 2002. INDISIM, an individual-based discrete simulation model to study bacterial cultures. J. Theor. Biol. 214, 305–319. Ginovart, M., Xifr´e, J., L´opez, D., Silbert, M. Individual based modelling of yeast growth in batch cultures, submitted for publication. Hsu, J.W.C., Speers, R.A., Paulson, A.T., 2001. Modeling of orthokinetic flocculation of Saccharomyces cerevisiae. Biophys. Chem. 94, 47–58. Ngondi-Ekome, J., Thiebault, F., Strub, J.M., van Dorsselaer, A., Bonaly, R., Contino-Pepin, C., Wathier, M., Pucci, B., Coulon, J., 2003. Study on agglutinating factors from flocculent Saccharomyces cerevisiae. Biochimie 85, 133–143. Powell, C.D., Quain, D.E., Smart, K.A., 2003. The impact of brewing yeast cell age on fermentation performance, attenuation and flocculation. FEMS Yeast Res. 3, 149. Stratford, M., 1992. Yeast flocculation: a new perspective. In: Rose, A.H. (Ed.), Advances in Microbial Physiology, vol. 33. Academic Press, London, pp. 1–71. Stratford, M., Coleman, H.P., Keenan, M.H.J., 1988. Yeast flocculation: a dynamic equilibrium. Yeast 4, 199–208. Stratford, M., Keenan, M.H.J., 1987. Yeast flocculation: kinetics and collision theory. Yeast 3, 201–206. van Hamersveld, E.H., van der Lans, R.G.J.M., Caulet, P.J.C., Luyben, K.C.A.M., 1998. Biotechnol. Bioeng. 57, 330–341. Verstrepen, K.J., Derdelinckx, G., Delvaux, F.R., 2003. Yeast flocculation: what brewers should know. Appl. Microbiol. Biotechnol. 61, 197–205. Walker, G.M., 1998. Yeast Physiology and Biotechnology. Wiley, Chichester.
Flocculation in brewing yeasts: A computer simulation study M. Ginovart a,∗ , D. L´opez b , A. Gir´o b , M. Silbert c,d a b
Departament de Matem`atica Aplicada III, Edifici ESAB, Universitat Polit`ecnica de Catalunya, Campus del Baix Llobregat, Avda. Canal Ol´ımpic, 08860-Castelldefels, Barcelona, Spain Departament de F´ısica i Enginyeria Nuclear, Edifici ESAB, Universitat Polit`ecnica de Catalunya, Campus del Baix Llobregat, Avda. Canal Ol´ımpic, 08860-Castelldefels, Barcelona, Spain c Departamento de F´ısica, Universidad de Extremadura, 06071 Badajoz, Spain d Institute of Food Research, Norwich Research Park, Colney, Norwich NR4 7UA, UK Received 27 July 2005; received in revised form 14 August 2005; accepted 5 September 2005
Abstract A new simulator, INDISIM-FLOC, based on the individual-based simulator INDISIM, is used to examine the predictions of two different models of yeast flocculation. The first, proposed by Calleja [Calleja, G.B., 1987. Cell aggregation. In: Rose, A.H., Harrison, J.S. (Eds.), The Yeasts, vol. 2. second ed., Academic Press, London, pp. 165–238 (Chapter 7)], is known as the “addition” model. The second, proposed by Stratford [Stratford, M., 1992. Yeast flocculation: a new perspective. In: Rose, A.H. (Ed.), Advances in Microbial Physiology, vol. 33. Academic Press, London, pp. 1–71] is known as the “cascade” model. The simulations show that the latter exhibits a better qualitative agreement with available experimental data. © 2005 Elsevier Ireland Ltd. All rights reserved. Keywords: Discrete simulation; Yeast flocculation; Individual-based model; Addition model; Cascade model
1. Introduction Yeast flocculation is the phenomenon of cellular aggregation when cells adhere, reversibly, to one another to form macroscopic flocs. It is an important process for the production of beer that causes the yeast to sediment, or cream, at the end of the fermentation process. Thus, the yeast can be harvested from the bottom (lager fermentation) or top (ale fermentation) of the fermenter and used for the next fermentation. Yeast flocculation remains an active field of research (for a recent, short, review see Verstrepen et al., 2003). Whilst there appear to be similarities with the process of sedimentation, or creaming, in colloidal disper-
∗
Corresponding author. Tel.: +34 93 5521133; fax: +34 93 5521001. E-mail address: [email protected] (M. Ginovart).
sions, the mechanisms involved are different (Stratford, 1992). In fact, yeast flocculation is a very complex process that depends on the expression of specific flocculation genes (Bony et al., 1998). Turbidimetric tests suggest that flocculation in bottom fermenting, but not necessarily in top fermenting, strains is mediated by a lectin-aggregating mechanism (Dengis et al., 1995; Ngondi-Ekome et al., 2003). Actually, yeast flocculation is a process in which biological, chemical and physical processes are involved. In this work, we are concerned mainly with the latter. In general, the flocculation of brewing yeasts is associated with the onset of the stationary phase, but the timing and degree of flocculation is not regulated precisely. This is an important consideration in brewing practice because if flocculation occurs too early, fermentation will cease prematurely, leaving residual sugar in the wort. This can cause microbiological stability problems and
0303-2647/$ – see front matter © 2005 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2005.09.002
52
M. Ginovart et al. / BioSystems 83 (2006) 51–55
can adversely affect flavour characteristics of the final product. Conversely, if it takes an unduly long time for cells to flocculate, this can cause downstream processing problems in beer clarification; see, e.g., Stratford (1992); Walker (1998). Flocculation may take place as a result of natural (in unstirred vessel) or external (mechanical) agitation. In this work, we are only concerned with the latter. It has been shown experimentally that flocculation requires an agitation threshold, namely a minimum kinetic energy to overcome the electrostatic, steric or hydrophobic repulsion between yeast cells or cell aggregates (Stratford and Keenan, 1987). After flocculation is started, the process advances rapidly to a new stationary state, namely a dynamical equilibrium between the rate of formation of new flocs and the rate of floc dissociation. It has also been observed that the size distribution of the flocs depends on the intensity of agitation. On the one hand the intensity of agitation favours the formation and growth of the flocs, while on the other it also contributes to their rupture through shearing (Stratford et al., 1988). Basically, in this steady state the system contains two fractions: flocs and single cells. We are concerned here with the simulation of the flocculation processes of a model yeast Saccharomyces cerevisiae using a new individual-based computer simulation code, INDISIM-FLOC, based on the generic simulator INDISIM1 (Ginovart et al., 2002), which we have developed to study bacterial growth. We are interested specifically with the consequences of the frequency of floc–floc collisions in the association/dissociation rates of flocs. As a result, the simulation of flocculation requires important changes to the computer code used in our previous yeast modelling studies (Ginovart et al., submitted for publication). The simulation of yeast growth and flocculation involve two different length and time scales, compared to those used in our previous studies of yeast growth, which have to be treated separately. Hence, whereas in the simulation of yeast growth the code is capable of controlling the characteristics of each cell, in the simulation of flocs it is only possible to control the number of “cells” in each floc. This, in turn, demands a new computer code, which we call INDISIM-FLOC, which will be described in Section 2. The changes in scale require that one single unit is made up of about O (102 ) individual yeast cells, which we henceforth call an “individual cell”, and that the unit of time is much larger than colli1
INDISIM is not publicly available at present. Please contact the corresponding author for specific information about our simulator and its applications.
Fig. 1. Experimental data (after Brohan and McLoughlin, 1984). Size distribution of the flocs as a function of their magnitude of the brewing yeast S. cerevisiae: () 250 rpm; (䊉) 500 rpm; () 750 rpm; () 1000 rpm.
sion times. These changes in scaling, when dealing with physical systems, are known as coarse graining, a term that we use here as well. Several recent theoretical models have been put forward to study different aspects of yeast flocculation by means of continuous models; see, e.g., van Hamersveld et al. (1998); Hsu et al. (2001). In this work, we examine, by means of computer simulations, two earlier models of yeast flocculation. The first, proposed by Calleja (1987), suggests that clustering is an additive process (addition model); the other, put forward by Stratford (1992), suggests a cascade process yielding fractal structures. We compare the results of our simulations, for each of the proposed mechanisms with the experimental results for one strain of the yeast S. cerevisiae (CIIIF) reported by Brohan and McLoughlin (1984), shown in Fig. 1; specifically the distribution of floc sizes as a function of the intensity of the agitation, and also with the experimental results reported by Stratford et al. (1988), for the strain S646-1B, specifically their Fig. 4 (not shown here) describing the effect of agitation on the number of cells within flocs. We discuss the experimental data in conjunction with our simulation results in Section 3. In fact, we suggest that our simulations may be regarded as com-
M. Ginovart et al. / BioSystems 83 (2006) 51–55
puter “experiments” with which we examine – within the set of rules used in the simulations – the predictions of the two models vis a` vis experiment. Our simulations are carried for a system of “elements” that includes both free individual cells and flocs. We define a floc as an aggregate of individual cells; our simulator controls each constitutive individual cell. The elements are placed in a cubic box which, in turn, is divided into small cubic spatial cells. The elements are allowed to move both within and between the small cubic cells. The simulator is set up so that the elements satisfy given rules of motion and of interaction. In fact, the simulator includes rules for the external agitation, the random motion of the elements and their collision. The following two comments are in order. First, within this approach, differentiation between different strains of the same yeast or between different types of yeast is limited to size and shape of the individual cells, as well as the strength and functional form of the aggregation and/or dissociation probabilities. Hence our conclusions regarding the two proposed mechanisms are generic rather than specific. Second, it is assumed that the simulated yeast is flocculent. In the next section, we report briefly on the main new features of the simulator used for this study. Then we present, and briefly discuss, some of the results of our simulations. 2. The simulator We assume an initial population of individual cells towards when the flocculation process starts, and such that: (1) The external (mechanical) agitation A to which the system is subjected is characterized by a parameter that we assume to be constant throughout the simulation. This parameter may be assumed to be the necessary kinetic energy contribution to overcome the repulsive interactions between the flocs and conditions the behaviour of the system. Whence A ∝ kinetic energy ∝ w2 . A is fixed at the beginning of the simulations and it will determine the probabilities of aggregation or dissociation of flocs. (2) The simulator controls flocs of various sizes containing n cells, including individual cells (n = 1). Henceforth the term “floc” also includes the case of an individual cell. Flocs are assumed to exhibit spherical symmetry, and are made up with n, closely packed, individual cells. The starting point for the simulations is an initial configuration of single cells randomly distributed in the spatial domain.
53
(3) The spatial domain is a cube subdivided into cubic cells, subject to periodic boundary conditions. Each spatial cell may contain one, several or no flocs. Our simulator controls each constitutive individual cell. Time is divided into equal intervals associated to computer steps. (4) At each time step, the simulator controls the size (n) of each floc and its position in space, characterized by a vector. Each floc is displaced to a new position randomly chosen within a given range R. We assume for this work that this range has a constant value, R = k. There are other possibilities that will be explored in another work, namely (i) that the range is a linear function of the rate of agitation, R = k A or (ii) that the range depends not only on A but also on the mass of the floc involved, say R = k (A/m)1/2 . We intend to probe into these possibilities and will report our simulation results on completion. (5) When a floc moves into a new small cubic cell, the simulator checks whether the spatial cell is occupied with other flocs. If so, one is chosen randomly; the simulator assumes that the two have collided and proceeds to estimate whether the two flocs aggregate or dissociate. (6) In order to compare between the two flocculation models (addition and cascade) we assume in the case of the cascade model, but not in the case of the addition model, that the colliding flocs are of similar size. (7) The following two probabilities are evaluated: (a) aggregation probability Pa and (b) dissociation probability Pd . In both addition and cascade models, the simulator assumes Pa to be constant, and the dissociation probability Pd is related to the exponential of A (the exponential of kinetic energy), and to the size n of the floc. If there is dissociation the distribution of sizes of the two resulting flocs are determined randomly. (8) At each “collision” the simulator evaluates sequentially: (a) Pa ; (b) if there is aggregation, the simulator evaluates the Pd of the resulting floc; (c) if there is no aggregation, the simulator evaluates separately Pd for the two “colliding” flocs. (9) As implied in (5), the constituents of the system do not remain static in their spatial cells. Here, we assume that their motion is solely due to the external agitation. 3. Results and discussion We are concerned here with the relationship between the intensity of agitation and the size distribution of flocs
54
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Fig. 2. Simulation results showing the effects of the agitation on the size of the flocs: (a) cascade model and (b) addition model.
and, with our simulator, we have examined two different mechanisms: the addition and the cascade models. We start with Fig. 2, which shows the simulation results for the number of individual cells per floc as a function of agitation. We find that this number diminishes with increasing agitation, in agreement with experiment (Fig. 4 in Stratford et al., 1988). Between 110 and 140 rpm, the experimental data exhibit an abrupt fall in floc size. This sharp fall appears in the simulations for the cascade model (Fig. 2a), but not in the addition model where floc size decreases continuously with agitation (Fig. 2b). There is, however, an important difference in the abrupt fall of floc size between experiment and simulation of the cascade model. In the latter, we find two abrupt falls; one at A1/2 ∼ 1.5 (corresponding to, approximately 144 rpm) and an even sharper fall at A1/2 ∼ 3 (≈617 rpm). Whichever way, the comparison between experimental data and simulations appears to favour the cascade model. We now turn to Fig. 1 which shows the experimental data of Brohan and McLoughlin (1984) displaying the effect of agitation on one of the distributions of floc size of the yeast S. cerevisiae (CIIIF). At the lower agitation rates (250 and 500 rpm), there is a wider distribution of floc sizes, with the lower containing a relatively larger percentage of large flocs. At the higher agitation rates (750 and 1000 rpm), the distribution of floc sizes is more limited; the narrower distribution corresponding to 1000 rpm. Note that only the points are experimental data; the lines joining them are guides to the eye. The authors do not include error bars in their plots. However, assuming that similar errors to those shown in Fig. 4 of Stratford et al. (1988) have been made in estimating floc sizes, then we expect these errors to be larger at the
higher agitation rates. Whence the error bars are probably more relevant in those cases when the distribution of floc sizes is narrower. Our simulations are intrinsically exact. Error bars in our simulations may be obtained by repeating the simulations with different initial configurations; this was not done for this work, but are assumed to be very small. We now turn to the simulations, shown in Fig. 3. Since we originally carried out our simulations in terms of A1/2 we used those simulation values, in rpm, nearest to the experimental data. Fig. 3a shows the results of our simulation using the cascade model, namely when only flocs of similar sizes are allowed to collide; while Fig. 3b shows the results of our simulations using the addition model. The cascade model simulations, Fig. 3a, shows
Fig. 3. Size distribution of the flocs as a function of their magnitude of the brewing yeast S. cerevisiae from the simulations: () 310 rpm; (䊉) 481 rpm; () 723 rpm; () 1012 rpm. On the left with the cascade model and on the right with the addition model.
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a wider range of size distributions than the experimental data, at both 250 and 500 rpm. On the other hand, for the same agitation values, the size distribution in the addition model, Fig. 3b, is narrower than the experimental data, even allowing for error bars similar to those indicated in Stratford et al. (1988). Moreover, and also in agreement with experiment, the distribution of larger flocs at 250 rpm in the simulations of the cascade model, is higher than for 500 rpm. Except for one value this is not the case with the addition model. The comparison between Fig. 1 and Fig. 3(a and b) again suggests that the cascade model exhibits a better qualitative agreement with the experimental data. We conclude that our simulations validate the cascade model, within the rules used in the simulations reported above. Finally, the following comment is in order. Our initial population does not consider the distribution of genealogical ages which, in our simulations, may be modelled by a distribution of “scars” on the surface of individual yeast cells (Ginovart et al., submitted for publication). Moreover, there is an intrinsic difficulty in relating the distribution of scars in yeast cells with the assumptions that need to be made in order to say something about the distribution of genealogical ages in individual cells (which contain O (102 ) yeast cells). There is, however, recent evidence that aged cells display an enhanced flocculation potential (Powell et al., 2003). We hope to be able to surmount the difficulties involved to incorporate in our simulator sensible assumptions that allow us to take the individual cell average age into account. There is also the added difficulty as to how we include the extra degree of “stickiness” in individual cells that, on average are deemed as “aged”, and what would be the case when a “young” and an “aged” individual cells collide. We are studying this problem at present and, if we succeed in overcoming these difficulties, will report our results on completion. Acknowledgements We thank J. Valls for valuable discussions and their general interest in this work. One of us (MS) has been supported by a Marie Curie Fellowship of the IHP European Community Programme under Contract no. MCFI-
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2002-00081. This work has been partially supported by DURSI, Generalitat of Catalonia (2003ACES0064) and by Ministry of Education and Science of Spain (DGYCIT PB97-0693; GL2004-01144). We are grateful to the anonymous referee for critical comments and suggestions that, we hope, have resulted in an improved manuscript. References Bony, M., Barre, P., Blondin, B., 1998. Distribution of the flocculation protein, Flop, at the cell surface during yeast growth: the availability of flop determines the flocculation level. Yeast 14, 25– 35. Brohan, B., McLoughlin, A.J., 1984. Characterization of the physical properties of yeast flocs. Appl. Microbiol. Biotechnol. 20, 16– 22. Calleja, G.B., 1987. Cell aggregation. In: Rose, A.H., Harrison, J.S. (Eds.), The Yeasts, vol. 2, second ed. Academic Press, London, pp. 165–238 (Chapter 7). Dengis, P.B., Nelissen, L.R., Rouxhet, P.G., 1995. Mechanisms of yeast flocculation comparison of top fermenting and bottom fermenting strains. Appl. Environ. Microbiol. 61, 718–728. Ginovart, M., L´opez, D., Valls, J., 2002. INDISIM, an individual-based discrete simulation model to study bacterial cultures. J. Theor. Biol. 214, 305–319. Ginovart, M., Xifr´e, J., L´opez, D., Silbert, M. Individual based modelling of yeast growth in batch cultures, submitted for publication. Hsu, J.W.C., Speers, R.A., Paulson, A.T., 2001. Modeling of orthokinetic flocculation of Saccharomyces cerevisiae. Biophys. Chem. 94, 47–58. Ngondi-Ekome, J., Thiebault, F., Strub, J.M., van Dorsselaer, A., Bonaly, R., Contino-Pepin, C., Wathier, M., Pucci, B., Coulon, J., 2003. Study on agglutinating factors from flocculent Saccharomyces cerevisiae. Biochimie 85, 133–143. Powell, C.D., Quain, D.E., Smart, K.A., 2003. The impact of brewing yeast cell age on fermentation performance, attenuation and flocculation. FEMS Yeast Res. 3, 149. Stratford, M., 1992. Yeast flocculation: a new perspective. In: Rose, A.H. (Ed.), Advances in Microbial Physiology, vol. 33. Academic Press, London, pp. 1–71. Stratford, M., Coleman, H.P., Keenan, M.H.J., 1988. Yeast flocculation: a dynamic equilibrium. Yeast 4, 199–208. Stratford, M., Keenan, M.H.J., 1987. Yeast flocculation: kinetics and collision theory. Yeast 3, 201–206. van Hamersveld, E.H., van der Lans, R.G.J.M., Caulet, P.J.C., Luyben, K.C.A.M., 1998. Biotechnol. Bioeng. 57, 330–341. Verstrepen, K.J., Derdelinckx, G., Delvaux, F.R., 2003. Yeast flocculation: what brewers should know. Appl. Microbiol. Biotechnol. 61, 197–205. Walker, G.M., 1998. Yeast Physiology and Biotechnology. Wiley, Chichester.