Applying Membrane Systems in Food Engineering

Applying Membrane Systems in Food Engineering Gabi Escuela1, Thomas Hinze2, Peter Dittrich2, Stefan Schuster2, Mario Moreno-´ Alvarez3 1Dept. of Mathematics and Computer Science, Bio Systems Analysis Group Friedrich Schiller University Jena, Germany [email protected] 2School of Biology and Pharmacy, Dept. of Bioinformatics Friedrich Schiller University Jena, Germany 3Lab. de Biomol´eculas, Universidad Sim´on Rodr´ıguez Canoabo, Venezuela Summary. Food engineering deals with manufacturing, packaging and distributing systems for drug and food products. In this work, we discuss about the applicability of membrane systems to model environmental conditions and their effects on the produces during storage of fresh fruits and vegetables. In particular, we are interested in abstract molecular interactions that occur between produce, film and surrounding atmosphere factors involved in fresh fruit and vegetable package designs. We present a basic implementation to simulate the dynamical behaviour of these systems, due to gas exchanges and temperature fluctuations. Additionally, we reveal the benefits of this modelling approach and suggest some extensions as future directions to be considered. 1 Introduction Membrane systems [17], also called P systems, had emerged to assist in the modelling of systems of concurrent reactions taking place in compartments, so as occur in biological systems. In this paper we use P systems as membrane structures delimiting compartments that contain multisets of objects representing molecules. The model was first presented in [3]. Compartments configuration changes over time (evolves) according to given rules that represent biochemical reactions and diffusions. In contrast to ODE-based approaches, each single molecule within the entire system is represented explicitly as individual entity. Capturing aspects of structural dynamics (changes in the membrane structure as well as in the composition of complex molecules) is seen as an advantageous feature of P systems. Inclusion of reaction kinetics into this formalism can be done by discretised kinetic laws [10]. We applied this mathematical formalism to a real known problem in fruits and vegetables post-harvest processing. 110 G. Escuela et al. Fresh fruits and vegetables are living materials that continue to respire after harvesting exhibiting progressive biochemical changes. Food engineering methods to preserve freshness of post-harvest produces include low temperature storage and special packaging technologies, mainly Modified Atmosphere Packaging (MAP). MAP of fresh fruits and vegetables refers to the technique of enveloping the produce in a sealed container of polymeric film in order to modify the O2and CO2 concentrations inside the package, reducing metabolic activity and increasing shelf life [16]. Designing MAP systems is a complex task that involves considerations about many interrelated environmental (as temperature and atmosphere composition), biological and package technology factors. Basic biological processes are respiration, transpiration, ethylene production and compositional changes due to metabolism. The variability of responses to internal and external signals depends on the characteristic of each plant organ type, developmental stage and physiological condition. In addition, much of the behaviour of a MAP system at cellular level are not fully understood. As examples we can refer to the little knowledge about the effect of CO2on the activity of respiratory enzymes [11]. Moreover, the contribution of the biochemical changes that alters physical properties of cell walls and tissues modifying the texture of the produce is not known in detail [9]. On the other hand, the mechanism of ethylene signal transduction that coordinates fruit ripening processes, is another aspect subject to study [1]. The difficulty to test different combination of gases and temperatures and the complexity of experimental setup for MAP systems had led to the development of various mathematical models [4, 11, 16, 19] and software [14]. In the literature, many respiration models are empirical fits of experimental data, based on one particular type and variety of fruit or vegetable, and most of them are based on the principles of enzyme kinetics and are represented using ODEs (for reviews see [5, 18]). However, there exists some lack on studies about the dynamical behaviour of these systems in terms of changes in environmental conditions, so as produce composition and physiology due to developmental processes [5]. It is worth mentioning that post-harvested fruits and vegetables, unlike other living materials, can be considered as less robust systems, as their responses on environmental fluctuations depends mostly on their actual configuration of biochemical components. In this context, some authors [7, 14] have considered the potential benefits of a systematic analysis or process-based modelling approach for fruits and vegetables. Considering that understanding the reaction network underlying MAP systems can give food experts more knowledge about emergent properties of packaged fruits and vegetables, we propose a framework based on membrane systems that abstracts basic biochemical reactions that occur in MAP systems. In the future, the proposed model can serve as a predictive tool to simulate changes in fresh produce on the molecular level, due to changes in environmental conditions. This paper is organized as follows: firstly, biological and technical background of MAP systems is described in section 2. In Section 3 we present a P system framework for MAP, including the description of components, reaction kinetics Applying Membrane Systems in Food Engineering 111 and evolution of the system. Section 4 shows an application of the framework considering a package under modified atmosphere containing two produces. Finally, in Section 5 we point out some benefits of using our framework and future extensions of it. 2 Fresh Fruits and Vegetables Packaging From the horticultural point of view, there are four main, sometimes overlapped, developmental stages identified in fruits, vegetables and flowers: growth, maturation, ripening and senescence. For packaging technology purposes, maturation and ripening stages of fresh commodities constitutes the central point of attention. During ripening, fruits and vegetables suffer considerable physiological changes that normally include: modification of colour through the alteration of chlorophyll, carotenoid, and/or flavonoid accumulation; textural modification via alteration of cell turgor and cell wall structure and/or metabolism; modifications of sugars, acids, and volatile profiles that affect nutritional quality, flavour, and aroma; and, generally enhanced susceptibility to opportunistic pathogens (likely associated with loss of cell wall integrity) [8]. The climacteric is a stage of fruit ripening associated with an autocatalytic production of ethylene, that rises cell respiration in some fruits. This physiological process marks the end of fruit ripening and the beginning of fruit senescence, in which a serie of irreversible events leads to breakdown and death of the plant cells. In addition to temperature control, a reduced O2and elevated CO2atmo- sphere can extend the post-harvest life of whole and pre-cut commodities. This techniques reduce their respiration rate as well as production of ethylene, minimizing metabolic activity, delaying enzymatic browning and retaining visual appearance [13]. In order to obtain a good design of such a system, it is necessary to understand many concurrent reactions at a macro, meso and micro level, including the dynamics behind the interdependencies between environmental, biological and technical factors. Basic environmental factors to be consider in MAP design are temperature, relative humidity (RH), and atmosphere composition. Other minor factors are light, chemicals as fungicides, growth regulators, ethylene perception blockers, etc. Temperature is the most important extrinsic factor affecting all elements of harvested produce. Considering packaged produces, temperature influences both the gas exchange of the produce and the permeability of the film for O2,CO2and H2O. High RH in the atmosphere surrounding fruit diminishes dehydration and preserves freshness, whereas excessive RH may engender moisture condensation, microbial growth and decay of the produce. On the other hand, three gases in the surrounding atmosphere O2,CO2and ethylene (C2H4), mostly influence stored fresh fruits and vegetables. Decreasing oxygen partial pressure can increase shelf life, but it is essential that the oxygen level not be reduced to the point that anaerobic respiration occurs. Anaerobiosis results in fermentation, the chemical conversion of 112 G. Escuela et al. carbohydrates into ethanol and organic acids, which may cause undesirable odours and flavours. Ethylene is a gaseous plant hormone (signal molecule) that regulates fruit ripening and senescence. O2is required for the synthesis of ethylene, while O2and low levels of CO2are required for its biological activity [9]. The most important biological processes occurring in fresh fruits and vegetables that affect them in packaging conditions over time are respiration, transpiration, ethylene production and compositional changes due to metabolism. Additionally, developmental processes, so as physiological and pathological breakdown should be considered. Respiration involves a complex set of biochemical processes of which part is the oxidative breakdown of carbohydrates, lipids and organic acids into CO2and H2Oplus heat and metabolic energy. Respiration rate can be expressed in terms of O2consumed or CO2produced. The respiratory quotient (RQ), the ratio of CO2produced to O2consumed, ranges from about 0.7 to 1.4 depending on the substrate and its metabolic state (if the substrate is a lipid, RQ < 1, and RQ > 1 for organic acids)[5]. When carbohydrates are aerobically respired, the RQ is near 1, and the reaction is represented by Eq. (1). C6H12O6+ 6 O2→6CO2+ 6 H2O+energy (1) On the other hand, post-harvested fresh fruit and vegetables are mainly made up of water (80 to 95% approx.). Water loss is the primary cause of fresh weight loss and it is much more sensitive to changes in relative humidity around the commodity than to the rate of respiration[9]. Transpiration occurs due to the fact that fruits and vegetables internal atmosphere is saturated with water vapour, while external atmosphere contains lesser. Therefore, water loss rate depends on the external and internal water vapour pressure gradient. There are two types of MAP techniques: passive or active. Passive MAP or equilibrium modified atmosphere consists in matching film permeation rates for O2and CO2with the respiration rate of the packaged produce. In some cases, it is likely that atmospheres within MAP will be actively established and adjusted, and this is realised creating a vacuum into the package and replacing the atmosphere with the desired gas mixture, and/or introducing gas absorbers/emmitters or other atmosphere-modifying elements into the package, so as using specialized films [18]. In a MAP design process, the type of material and its surface area and thickness are selected to obtain the desired equilibrium gas composition. Each film type has specific ranges of O2and CO2permeabilities, usually with the permeability of CO2being 3 −5 times that of O2[4]. Two strategies for creating film barriers exist: continuous and perforated films. Continuous film control movement of O2 and CO2into or out of the package so that steady-state O2and CO2levels are achieved in the package, that is, they are used in the MAP design process assuming a constant respiratory rate of the produce. Perforated films with small holes are more suitable for produce having a high O2demand, and in this case the rate of gas exchange is a sum of gas permeation through the film and gas diffusion through the microperforations [9]. Applying Membrane Systems in Food Engineering 113 3 A P System-based MAP We abstract a fruit or vegetable as a graph of cells or modules, like tissue P systems [15]. Each cell represents a compartment that contains species, and at a specific time, the contents of the compartment determines the cell configuration. This serves as a mechanism to differentiate one cell from other, given the possibility to create diverse tissue types, as occurs for example in fruits epicarp, mesocarp and endocarp tissues [7]. Additionally, as gas consumption-production occurs inside the cells, at the mitochondria level, and is stated that gas diffusion between cells depends on the geometry of the produce [11], differences in gas content in cells that conform a determinate region can adequately be represented. This is also in accordance to the idea that the ripening process usually starts in one region of a fruit and spreads to neighbouring regions, due to ethylene diffusion starting from promoter cells [1]. Produces into the package are represented as a population of membranes, giving the advantage that the model can deal with distinct fruits and vegetables within the same film, or the same produce in distinct developmental stages, varieties and/or presentations. Figure 1 shows as example, the schematic representation for such a system. In the next section we present the formal specification of our model. Fig. 1. A schematic representation for the MAP system model. In this case, two produces share a package: plant1is formed by three connected cells, and plant2is formed by a single cell. Arrows represent paths for molecules (spheres) diffusions Multiset prerequisites Let Abe an arbitrary set and Nthe set of natural numbers including zero. A multiset over Ais a mapping F:A−→ N∪ {∞}.F(a), also denoted as [a]F, specifies the multiplicity of a∈Ain F. Multisets can be written as an elementwise enumeration of the form {(a1, F (a1)),(a2, F(a2)), . . .}since ∀(a, b1),(a, b2)∈F: b1=b2. The support supp(F)⊆Aof Fis defined by supp(F) = {a∈A|F(a)> 0}. A multiset Fover Ais said to be empty iff ∀a∈A:F(a) = 0. The cardinality 114 G. Escuela et al. |F|of Fover Ais |F|=Pa∈AF(a). Let F1and F2be multisets over A.F1is a subset of F2, denoted as F1⊆F2, iff ∀a∈A: (F1(a)≤F2(a)). Multisets F1and F2are equal iff F1⊆F2∧F2⊆F1. The intersection F1∩F2={(a, F (a)) |a∈ A∧F(a) = min(F1(a), F2(a))}, the multiset sum F1]F2={(a, F(a)) |a∈ A∧F(a) = F1(a) + F2(a)}, and the multiset difference F1F2={(a, F (a)) |a∈ A∧F(a) = max(F1(a)−F2(a),0)}form multiset operations. Multiplication of a multiset F={(a, F(a)) |a∈A}with a scalar c, denoted c ·F, is defined by {(a, c·F(a)) |a∈A}. P system components Let N+=N\ {0}be the set of natural numbers without zero, and m, n ∈N+. We define a P system for a MAP system as a construct: ΠMAP = (µ, S, plant1, . . . , plantm, G, L0, D1, . . . , Dd, f1, . . . , fd, ∆τ) where: •µ=[[[]cell1,1...[]cell1,n1]plant1...[[]cellm,1...[]cellm,nm]plantm]package is the spatial system structure composed of three inner levels: package, plants, and cells, •Sis a set of chemical species, •plant1, . . . , plantmrepresent the produces into the package, •Gis a set of global parameters, •L0:S→Nis a multiset of axioms representing the initial molecular configuration, •Dνis a diffusion (communication) rule among package and external environment (ν= 1, . . . , d), •fν: (S→N)→Nis a kinetic function attached to diffusion rule Dν, •∆τ ∈R+is the time discretisation interval. A diffusion rule Dνcan be of the form [s]→[]sfor molecules s∈Sleaving the package and released to the external environment, and []s→[s] for molecules entering the package, respectively. Furthermore, each plantiis defined as a tuple: planti= (Ni, Ei, Gi, Di,1, . . . , Di,di, fi,1, . . . , fi,di) where: •Ni={celli,1, . . . , celli,ni}defines a set of cells within plant i, •Ei⊆Ni×Nispecifies a set of directed edges (diffusion channels between cells), •Giis a set of plant (organ) specific parameters, •Di,κ represents a diffusion rule inside plant iand between plant iand package (κ= 1, . . . , di), •fi,κ is a kinetic function attached to diffusion rule Di,κ. Applying Membrane Systems in Food Engineering 115 Here, a diffusion rule can be of the form [s]cellp,q →[]cellp,q sfor molecules s∈S leaving cellp,q and spread out into the package. A rule of the form []cellp,q s→ [s]cellp,q describes molecules entering cellp,q from the package. Finally, a rule of the form [s]cellp,q →[s]cellx,y formulates the directed transport of molecule salong the edge (cellp,q, cellx,y)∈Ei. Each celli,j is defined as a tuple celli,j = (Li,j,0, Ri,j,1, . . . , Ri,j,ri,j , fi,j,1, . . . , fi,j,ri,j ) where: •Li,j,0:S→Nis a multiset of axioms representing its initial molecular configuration, •Ri,j,k = (Ai,j,k, Bi,j,k) with Ai,j,k :S→N(multiset of reactants) and Bi,j,k :S→N(multiset of products) specifies a reaction rule including its stoichiometric factors, •fi,j,k : (S→N)→Nis a function corresponding to kinetics of reaction Ri,j,k. System evolution A P system of the form ΠMAP evolves by successive progression of its configuration at discrete points in time t∈Nfor what we assume a global clock. Two consecutive dates tand t+ 1 specify a time span ∆τ. A system step at time tconsists of three modification stages carried out from outer to inner spatial components of the system. Firstly, the diffusion between package and its environment is considered. To this end, the rules D1up to Ddare employed. Afterwards, the diffusion between package and cells as well as the intracellular diffusion is utilised by employing the rules Di,κ for each plant i= 1, . . . , m. The last modification stage concerns application of the reaction rules specified in each cell. To cope with conflicts that can occur if the available amount of substrate cannot satisfy all matching diffusion and reaction rules, we prioritise all rules by their index: D1> D2> . . . > Dd. Moreover, for each plant i:Di,1> Di,2> . . . > Di,diand for each cell i, j: Ri,j,1> Ri,j,2> . . . > Ri,j,ri,j . Thus, we keep determinism of the system evolution and enable mass conservation. An alternative method for coping with conflicts is randomisation in selection and sequentialisation of diffusion and reaction rules. The application of an arbitrary rule is organised into two consecutive steps. The first step identifies all molecules from the rule’s left hand side acting as sources for diffusion or reactants. These molecules are removed from the current configurations. Corresponding molecules from the right hand side (destinations in case of diffusion and products in case of reactions) are then added. We formulate discretised reaction-diffusion kinetics by specification of scalar functions f:M→Nbased on a multiset M:S→N. Each function fconverts the current configuration (Ltor Li,j,t), a multiset of objects, into the number of turns for application of the corresponding diffusion or reaction rule. Here, kinetic laws ˆ f(s) for each species s∈Semploy the multiplicity of its occurrences to formulate the corresponding reaction rate. 116 G. Escuela et al. For updating the entire system configuration, we define an iteration scheme as shown in Figure 2. When this formalism is hidden in a software, the specification is intuitive and accessible for an expert focussing on MAP modelling. 4 Simulation As a first application, we introduced as rules into the model only the basic processes involved in a MAP design: respiration and fermentation, so as gas diffusion between membranes. The influence of gas composition on respiration rates of produce has been widely represented by Michaelis Menten-type equation [5]. In this context, respiration rate is considered as a function of concentration in terms of enzymatic reaction, with O2in the place of substrate and the product CO2acting as inhibitor. Temperature dependence over respiratory rate and over film permeability was represented using Arrhenius equation (Eq. 2). k=F×e−Ea/R×T(2) where Eais the activation energy, expressed in joule per mol, defined as the energy that must be overcome for a chemical reaction to occur; Ris the gas constant (≈8.314 J·K−1mol−1), Tthe absolute temperature, Fis the pre-exponential factor that represents the total number of molecular collisions per second; and k corresponds to the number of collisions per second that result in a reaction. This can be related to the probabilistic approach to P systems introduced by [2] in order to obtain more biological-like models. In this context, the Arrhenius exponential term can be viewed as the probability per time unit that the reaction takes place. In order to apply our model, we simulate the dynamical behaviour of an instance of a ΠMAP with two hypothetical fruits as it is shown in Fig. 1, using continuous film and passive MAP as package techniques. Rules that use symbol between reactants and products must been interpreted as reversible reactions. Into the formalism described in Fig. 2, a rule of the form Dα= [σ][]σ, for example, consists in the following two rules, in order of application: [σ]→[]σand []σ→[σ]. Temperature is represented by Tand expressed in Kelvin (K). O2,CO2and H2Oabundances in the outside are represented by O2ex,CO2ex and H2Oex respectively. Aand Esymbolise the surface area in cm2and the thickness of the packaging film in mil (1mil = 0.00254cm). pO2 and pCO2 represent the reference film permeability in mL ·mil ·cm2·hr−1·atm−1for O2,CO2and H2O, respectively. EaO2 and EaCO2 symbolise the permeability activation energy expressed in J·mol−1for O2and CO2, respectively. Misymbolises mass of the produce iin kg. For simplicity, we assume that each cell in a produce has the same mass. rO2 and rCO2fcorresponds to the preexponential factor for produce respiration and fermentation in mL ·kg−1·hr−1.ErO2 and ErCO2frepresent the respiration and fermentation activation energy for the produce expressed in J·mol−1. Applying Membrane Systems in Food Engineering 117 Stage 1 (diffusion between package and external environment): ∀α= 1,...,d diffusion rule Dαconditions action [σ]→[]σ(σ∈S)∧({(σ, fα)} ⊆ Lt)Lt:= Lt {(σ, fα)} []σ→[σ] (σ∈S)Lt:= Lt] {(σ, fα)} with fα(Lt) = jkα(G)·∆τ ·ˆ f(|Lt∩ {(σ, ∞)}|)k Stage 2 (diffusion between plant cells and package): ∀i= 1,...,m ∀α= 1,...,di diffusion rule Di,α conditions action [σ]celli,j →[]celli,j σ(σ∈S)∧(celli,j ∈Ni)∧ ({(σ, fi,α)} ⊆ Li,j,t) Li,j,t := Li,j,t  {(σ, fi,α)} Lt:= Lt] {(σ, fi,α)} []celli,j σ→[σ]celli,j (σ∈S)∧(celli,j ∈Ni)∧ ({(σ, fi,α)} ⊆ Lt) Lt:= Lt {(σ, fi,α)} Li,j,t := Li,j,t ] {(σ, fi,α)} [σ]celli,j →[σ]celli,k (σ∈S)∧(k6=j)∧(celli,j ∈Ni)∧ (celli,k ∈Ni)∧((celli,j , celli,k)∈Ei) ∧({(σ, fi,α)} ⊆ Li,j,t) Li,j,t := Li,j,t  {(σ, fi,α)} Li,k,t := Li,k,t ] {(σ, fi,α)} with fi,α(Lt) = jki,α(G, Gi)·∆τ ·ˆ f(|Lt∩ {(σ, ∞)}|)k Stage 3 (reactions occurring within each cell): ∀i= 1,...,m ∀j= 1,...,ni ∀α= 1,...,ri,j reaction rule Ri,j,α conditions action (Ai,j,α, Bi,j,α)fi,j,α ·Ai,j,α ⊆Li,j,t Li,j,t := Li,j,t fi,j,α ·Ai,j,α ]fi,j,α ·Bi,j,α with fi,j,α(Li,j,t) =          ki,j,α(G, Gi)·∆τ Q ∀c∈supp(Ai,j,α) : (Ri,j,α = (Ai,j,α, Bi,j,α)) ˆ f(|Li,j,t ∩ {(c, ∞)}|)|Ai,j,α∩{(c,∞)}|         Increment time t: Lt+1 := Lt ∀i= 1,...,m ∀j= 1,...,ni Li,j,t+1 := Li,j,t Fig. 2. Iteration scheme for the temporal evolution of ΠMAP system