Parameter Engineering vs. Parameter Tuning: the Case of Biochemical Coordination in MoK Stefano Mariani DISI, A LMA M ATER S TUDIORUM–Università di Bologna via Sacchi 3, 47521 Cesena, Italy Email: s.mariani@unibo.it Abstract—To cope with nowadays MAS complexity, nature- of such goals suitable for any kind of nature-inspired MAS inspired coordination models and languages gained increasing is possible. First of all, the fact that a given natural system attention: in particular, biochemical coordination models. Being works properly relying on a given set of parameters, each of intrinsically stochastic and self-organising, the effectiveness of which has a given set of functional dependencies with others, their outcome likely depends on a correct parameter tuning stage. doesn’t necessarily mean that the same sets of parameters In this paper, we focus on chemical reactions rates, showing that simply imitating chemistry “as it is” may be not enough and functional dependencies will work for an artificial system for the purpose of effectively engineer complex, self-organising drawing inspiration from the natural one. Then, to proficiently coordinated systems such as MoK . identify the relevant parameters and engineer their (possibly reciprocal) functional dependencies, a proper methodology is needed—which will likely rely on simulations. I. I NTRODUCTION Accordingly, the remainder of the paper is organized as Nowadays MAS are demanding new paradigms and ab- follows: Section II explains what biochemical coordination stractions to deal with their increasing complexity [1]. Such is (Subsection II-A) and reminds the importance of simula- complexity is mostly due to the number and nature of interac- tion tools for self-organising systems development (Subsec- tions happening within and between MAS [2], [3]. Coordina- tion II-B), also describing the MoK model (Subsection II-A1) tion models and languages – whose main goal is to govern as well as the BioPEPA tool (Subsection II-B1); Section III such interactions [4] – have historically drawn inspiration conveys the main contribution of the paper, introducing and from self-organising coordination in natural phenomena— motivating the notion of parameter engineering through a e.g., pheromone-based [5] and chemical [6] coordination. number of functional rates engineering examples; finally, Sec- Among the many, biochemical coordination has been shown tion IV concludes also giving some hints about further works. to be particularly effective [7], [8]. Here, there is no central authority ruling the interaction space. Instead, a number of local, stochastic coordination rules, to which all the interacting II. BACKGROUND agents “implicitly” obey – as they are the “laws of nature” – A. Biochemical Coordination drive MAS coordination. Thus, it happens by emergence as a consequence of the self-organisation between MAS coordi- The chemical metaphor appears particularly appealing for nated entities. Being a self-organising process, coordination MAS coordination due to the simplicity of its foundation effectiveness likely depends on a correct parameter tuning [6]. The idea is to coordinate any MAS entity (agents as stage, often performed in loop with a simulation stage [9]. well as information) as “molecules” floating in a chemical “solution”, whose evolution is driven by chemical “reactions” In the case of biochemical coordination, being the “laws continuously and spontaneously consuming and producing of nature” the (artificial) chemical reactions installed in the molecules. As many chemical reactions can occur at a given coordination medium, such parameters are, e.g., the rate of time, chemical solution evolution is driven by race conditions application of a chemical reaction, the concentration of the among their rates, which means certain reactions are stochas- chemicals participating the reaction and their stoichiometry— tically executing over others—as in chemistry actually is [10]. the “extent” to which chemicals participate. In this paper, we focus on rates, aiming at a twofold goal: Biochemical tuple spaces [7] enhance such metaphor by adding a spatial abstraction: the compartment. A compartment • on one hand, showing that the law of mass action is a tuple space equipped with biochemical reactions, driving for rate expressions may be not enough to effectively the evolution of the molecules floating in it. Compartments engineer a biochemical coordination middleware may be networked in “neighbourhoods” as in chemistry hap- • on the other hand, highlighting that designing arbitrary pens through membranes, so as to shape more complex spatial functional rates demands for a disciplined and prin- structures—such as tissues and organs. Computationally, bio- cipled approach different from “parameter tuning”, chemical tuple spaces are a stochastic extension of the L INDA which we call parameter engineering. model [11]: the idea is to equip each tuple with a “concentra- tion” value, representing a measure of the pertinency/activity In particular, such goals are defined w.r.t. the MoK model of the tuple (molecule) within the space (compartment)— and the BioPEPA tool, used as the subject and the means the higher it is, the more likely and frequently the tuple of investigation, respectively. Nevertheless, a generalisation will influence system coordination [7]. Such concentration is evolved by biochemical rules installed into the compartment, ◦ Aggregation3 — bounds together semantically affecting concentration values over time exactly in the same related molecules way chemical substances evolve into chemical solutions [10]— ◦ Reinforcement — consumes an enzyme to that is, according to the law of mass action [12], [10]. reinforce the related molecule The law of mass action is1 a mathematical model that enzyme(Molecule 1 ) + Molecule 1c 7−→rreinf explains and predicts the behaviour of solutions in dynamic Molecule 1c+1 equilibrium. It can be described with respect to two aspects: ◦ Decay — enforcing time situatedness, (i) the equilibrium aspect, concerning the composition of a molecules should fade away as time passes reaction mixture at equilibrium and (ii) the kinetic aspect, Molecule c 7−→rdecay Molecule c−1 concerning the rate equations for elementary reactions. The law states that the rate of an elementary reaction (rf ) – a ◦ Diffusion — space situatedness is inspired reaction that proceeds through only one transition state, that is by biology, therefore based upon diffusion to one mechanistic step – is proportional (kf ) to the product of neighbouring compartments (tuple spaces) the concentrations of the participating molecules (R1 , R2 ): {Molecule 1 Molecules 1 }σi + {Molecules 2 }σii 7−→rdiffusion S 1 2S rf = kf [R1 ][R2 ] {Molecules }σi + {Molecules Molecule 1 }σii kf is called rate constant and, in chemistry, is a function of These four biochemical reactions are the minimum set of participating molecules affinity—to learn more, please refer to coordination mechanisms believed (at the moment) to be [12] and therein cited bibliography. necessary and sufficient to properly drive a MoK -coordinated The MoK model, briefly described in next section, models MAS toward the desired behaviour regarding knowledge self- MAS coordinated entities as well as coordination processes organisation. Nevertheless, this set may be refined and ex- by (i) adopting the chemical metaphor abstractions and (ii) tended if its lack of expressiveness w.r.t. MoK desiderata be- borrowing (to some extent) from biochemical tuple spaces the comes evident. Anyway, having a well-defined set of primitives computational model. is a necessary step to start distinguishing what sorts of self- organising behaviours can and cannot be achieved with MoK . 1) The MoK Model: M olecules o f K nowledge [13] (MoK for short) is a model for knowledge self-organisation In fact, once such primitives are fixed, we can focus on the in MAS. The main goals of MoK are: issue of properly engineering their rate expressions—rreinf , rdecay , rdif f usion . In particular: is it sufficient to stick with the • to let information chunks autonomously aggregate into law of mass action to achieve MoK goals, or should we build heaps of knowledge our “custom” functional dependencies? If so, which parameters • to let knowledge autonomously flow toward the inter- and which kind of dependencies (direct, inverse, etc.) are to ested agents—rather than be searched be used in each rate expression? And how can MAS designers make such decisions? Here follows a brief summary of MoK model components— consider reading [13] and [14] for MoK formalisation and Section III answers these questions through a number of early application respectively: examples exploiting the BioPEPA simulation tool – briefly described in next section – to analyse different alternatives • MoK atoms — produced by a given source to convey regarding MoK reactions rate expressions. an “atomic piece of information”, atoms should also store some metadata to ease semantic characterisation B. Biochemical Simulation • MoK molecules — “heaps” for information aggrega- tion, they cluster together semantically related atoms Simulation has been widely recognized as a fundamental • MoK enzymes — enzymes reify knowledge-oriented development stage in the process of designing and implement- (inter-)actions made by agents and are meant to influ- ing both MAS as well as biochemical processes [16], [9]. This ence molecules’ concentration2 is mostly due to the high number of system parameters needed, the huge number of local interactions between components, the • FM oK function — as a knowledge-driven model, influence of randomness and probability on system evolution. MoK must have a way to determine the seman- A number of different simulation tools capable of modeling tic correlation between information, therefore, the biochemical-like processes exist, either born in the biochem- MoK function FM oK should be defined, taking two istry field (see [17] for a survey) or in the (Multi-)Agent Based molecules and returning a value m ∈ [0, 1]. Simulation research area (survey in [18]). Among the many, • MoK reactions — the behaviour of a MoK system is A LCHEMIST [19], PRISM [20], and BioPEPA [12] at least, determined by biochemical reactions, which stochas- are worth to be mentioned. Our choice fell on the latter for its tically – according to their rate – drive molecules appealing features – briefly described in next section – which aggregation, reinforcement, decay, and diffusion: perfectly suit the purpose of the paper. 1 From http://en.wikipedia.org/wiki/Law of mass action. 3 Aggregation reaction formalisation is not shown here because it has been 2 Please, notice that an atom is a “singleton molecule”, hence the term left out from BioPEPA simulations for the lack of expressiveness of the tool. “molecule” will be used also for “atom” from now on. To learn more, please refer to the technical report [15]. 1) The Bio-PEPA Tool: BioPEPA [12] is a language for a well-engineered approach—indeed, parameter engi- modeling and analysis of biochemical processes. It is based on neering prior to parameter tuning PEPA [21], a process algebra originally aimed at performance analysis of software systems, extending it to deal with some By generalisation, our first goal aims at showing there is the features of biochemical networks, such as stoichiometry and need to consider re-engineering natural system’s parameters, as different kinds of kinetic laws—including the law of mass well as their functional dependencies, so as to better cope with action. The most appealing features of BioPEPA are: the problem at hand—as done with other natural metaphors: most notably, the ACO approach to distributed optimisation, in • custom kinetic laws represented by means of func- which the original “ant” metaphor is indeed just a metaphor, tional rates not the actual implementation [22]. • definition of stoichiometry (“how many” molecules of Therefore, for each of the following experiments, we (i) a given kind participate) and role played by the species identify which are the desiderata for the MAS run-time be- (reactant, product, enzyme, . . . ) in a given reaction haviour, (ii) engineer rates by designing functional dependen- cies which are likely to pursue the chosen goal, (iii) include a • theoretical roots in CTMC semantics—behind any pure parameter tuning stage to fine-tune the MAS behaviour (if BioPEPA specification lies a stochastic labelled tran- needed). All of this is done one reaction (coordination policy) sition system modeling a CTMC at a time, thus one functional rate at a time, incrementally accumulated until composing the whole MAS behaviour. This In BioPEPA, rate expressions are defined as mathematical approach is what we call parameter engineering. equations involving reactants’ concentrations (denoted with the reactant name and dynamically computed at run-time) Furthermore, a principle we believe to be extremely im- and supporting mathematical operators (e.g. exp and log portant for engineering self-organsing systems will be kept in functions) as well as built-in kinetic laws (e.g. the law of mass mind: keeping the number of external parameters as small as action, denoted with the keyword fMA) and time dependency possible. For “external” we mean parameters which are ad hoc (through the variable time, changing value dynamically ac- added to the coordination model – MoK in our case – to better cording to the current simulation time step)4 . The BioPEPA design functional rate expressions—e.g., the law of mass action Eclipse plugin5 is the tool used in next section to investigate constant rate. On the contrary, internal parameters are those MoK reactions’ rates influence on system behaviour. already present in the coordinated MAS—e.g., in the case of MoK , the concentration of the reactants or the time flowing. III. PARAMETER E NGINEERING IN R ATE E XPRESSIONS The advantage of using internal parameters as opposed to external ones, lies in the fact that a system using more internal As far as nature-inspired MAS are concerned, simulation parameters than external ones is much more adaptive and self- tools are usually exploited to study how those parameters regulating, since it only relies on “within-system” information inherited from the natural metaphor influence the overall MAS rather than on “outside-system” data to dynamically adjust its behaviour. This is done with the aim to fine-tune such param- behaviour—in the case of MoK , reactions’ rates. eters value so as to get the better run-time “performances”— Technical Notes on Experiments: Each of the following whatever this means (often, a behaviour closer to that exhibited experiments has been performed by using Gillespie’s stochastic in nature). simulation algorithm in 30 independent replications. Each of But, what about the question of wether the natural system’s the following plots has been directly generated from BioPEPA parameters are well suited also for the artificial one? In as a result of the correspondent experiment—hence, of the 30 particular, w.r.t. biochemical coordination (thus, MoK also): Gillespie runs. In each chart, the x-axis plots the time steps of what about shaping our own rate expressions for biochemical the simulation, whereas the y-axis the concentration level of reactions rather than blindly relying on the law of mass action the reactants expressed in units of molecules. to define their functional dependencies? Do we gain any improvement w.r.t. the overall coordinated MAS behaviour? A. Injection Rate Furthermore: can the same improvement be achieved by simply fine-tuning the natural system’s parameters as they are in Althought injection of atoms into a MoK compartment is nature (e.g. the law of mass action constant rate)? not yet part of MoK ’s core set of formalised reactions, its influence on the system is so important to deserve its own Through the following experiments, we aim at answering analysis. Basically, injection can be described as follows: this kind of questions, hopefully achieving our twofold goal: Injection — Produces atoms out of sources without • showing that the law of mass action is too weak consuming them to effectively express a number of self-organising source(Molecule 1 ) + Molecule 1c 7−→rinj behaviours—such as MoK ’s source(Molecule 1 ) + Molecule 1c+1 • highlighting that shaping custom functional dependen- Two contrasting needs have to be addressed: on one hand, cies for rate expressions is a complex task demanding atoms should be perpetually injected into the MAS, since there 4 To learn more about BioPEPA syntax, please refer to [12]. is no way to know a-priori when some information will be 5 Instructions on how to install at http://homepages.inf.ed.ac.uk/s9552712/ useful; on the other hand, we would likely avoid flooding the bio-pepa/download.html, manual at http://homepages.inf.ed.ac.uk/stg/research/ system without any control on how many atoms are in play. biopepa/eclipse/manual/manual.pdf Thus, three options are viable: 1) make injection rates decreasing as time passes [14], e.g., option (3) is better, since in the news management 2) enforce some kind of “saturation” to stop injection scenario information (on average) loses relevance as time 3) a combination of the two passes. B. Decay Rate MoK decay reaction is an effective way to resemble the relationship between information relevance and time flow. Fur- thermore, decay enforces a kind of negative feedback which, together with the positive feedback provided by MoK en- zymes, enables the feedback loop peculiar of natural systems. Time dependency alone is not enough for a meaningful decay behaviour: by using, e.g., a fixed rate we end-up simply slowing down the saturation process provided by injection reaction. Hence, Fig. 3 shows three different combinations of time dependency and concentration dependency for MoK decay reaction—a fourth one (yellow line), based on the law of mass action, is given for comparison purpose: 1) linear time dependency + relative concentration depen- dency (blue dashed line) 2) logarithmic time dependency + relative concentration dependency (red line) Fig. 1. Comparison of functional rates for atoms injection. Horizontal lines represent correspondent sources’ concentration: purple dashed for option (1), 3) linear time dependency + built-in law of mass action pink for option (2), orange for option (3), light-blue for option (4). (green dashed line) Fig. 1 shows option (1) in blue, option (2) in yellow and option (3) in red. The green dashed line plots the law of mass action rate, whereas horizontal lines are the sources. Fig. 2 shows the BioPEPA functional rates specification used. 1 // option (1) 2 injE = [source_economics/atom_economics * (1 / (1 + time))]; 3 // option (2) 4 injS = [source_sports - atom_sports]; 5 // option (3) 6 injC = [(1 / (1 + time)) * (source_crime - atom_crime)]; 7 // option (4) 8 injP = [fMA(0.05)]; Fig. 2. The fMA keyword calls a built-in function to compute the law of mass action. Its only parameter is the rate constant. The fMA implicitly consider reactants involved in the reaction exploiting its correspondent functional rate— for the full BioPEPA specification, please refer to [15]. Clearly, using rate expressions based on the law of mass action is out of question: its behaviour follows none of MoK Fig. 3. Comparison of functional rates for atoms decay. Again, horizontal lines represent correspondent sources’ concentration: purple dashed for option injection reaction desiderata. Once discarded also option (1), (1), orange for option (2), light-blue for option (3), pink for option (4). whose trend is clearly too slow in reaching saturation, options (2) and (3) may seem almost identical. Actually they are not: Fig. 4 shows the BioPEPA functional rates specification used6 . • option (2) is “saturation-driven” only, thus if at some Again, the law of mass action is unsatisfactory, as well as op- point in time atom_sports will suddenly decrease tion (1). Options (2) and (3) are both viable solutions instead. in concentration – e.g. due to agents consuming them – The choice is mostly driven by how fast are the dynamics they will go back to saturation-level as fast as possible, of the scenario in which MoK has to be deployed, thus how no matter how long their sources are within the system fast information should lose relevance—e.g., in MoK -News, choice (2) has been preferred. Nevertheless, please notice that • option (3) instead, makes the saturation option (3) has an additional parameter w.r.t. option (2): the process time-dependant. In particular, the longer law of mass action “rate constant”. Furthermore, even if such source_sports are within the system, the slower parameter is made dynamic – e.g. the ratio between sources saturation will be and atoms concentrations as done in options (1), (2) – the Choosing among the two depends on the application-specific 6 Actually, the Heaviside function has been also used to counter BioPEPA context in which the MoK model is used. In MoK -News setting which allows rates to become negative—see [12]. 1 // option (1) 2 decayE = [source_economics / atom_economics * 3 time]; 4 // option (2) 5 decayC = [source_crime / atom_crime * 6 log(1+time)]; 7 // option (3) 8 decayP = [fMA(0.05) * time]; 9 // option (4) 10 decayS = [fMA(0.05)]; Fig. 4. BioPEPA specification of rate expressions for MoK decay reaction. trend still would not match our desiderata for MoK decay reaction—compare with yellow line of Figure 4 in [15]. C. Reinforcement Rate To properly engineer MoK reinforcement reaction rate, we have to keep in mind what enzymes are meant for, that is, (i) representing a situated interest manifested by an agent Fig. 6. Comparison of functional rates for atoms reinforcement. Lines worth w.r.t. a piece of knowledge – an atom or a molecule – to be considered are: the yellow one, plotting option (1), the dashed blue one, plotting option (2), the red one, plotting option (3). (ii) to be exploited to reinforce such knowledge “relevance” within the MAS. With the word “situated” we mean that reinforcement should take into account the situatedness of (red line) is doubled w.r.t. “yellow” enzymes (yellow line) in agents (inter-)actions along a number of dimensions: time, Fig. 6: as a result, the “duration” of the feedback is doubled as space, type—a “search” action, a “read” action, etc. For these well; (ii) in Fig. 8 the stoichiometry of “red” atoms (red line) in reasons, MoK reinforcement reaction rate should: reinforcement reaction is doubled w.r.t. “yellow” atoms (yellow • be prompt, that is rapidly increase molecules line) in Fig. 6: as a result, the “intensity” of the feedback is concentration—despite decay more than doubled. • limited both in time and space, to resemble relevance relationship with situatedness of (inter-)actions • depend on the (inter-)action type—e.g. a “read” action could inject more enzymes and/or reinforce atoms with greater stoichiometry w.r.t. a “search” action Fig. 6 clearly shows that our desiderata are fulfilled only by a reinforcement reaction having a functional dependency on the ratio between the reinforced molecule’s concentration and its source own—option (1) in Fig. 5. Once again, sticking 1 // option (1) 2 feedS = [(source_sports / atom_sports)]; 3 // option (2) 4 feedE = [fMA(source_economics / atom_economics)]; 5 // option (3) 6 feedC = [fMA(0.05)]; Fig. 5. BioPEPA specification of rate expressions for MoK reinforcement reaction. Fig. 7. Enzymes concentration increment effect on reinforcement. with the law of mass action alone is out of question: option (2) – dashed blue line –, even if adopting a dynamic rate Notice also: (i) Fig. 7 shows the opposite holds too, that constant, exhibits an exceedingly high and fast peak, option is, halving the initial concentration halves the duration of the (3) – red line –, using a fixed rate constant (as in the law feedback (yellow and blue lines); (ii) Fig. 8 shows that no of mass action typically is), almost completely ignores the interference happens between concentration and stoichiometry feedback—enzymes are too slowly consumed (orange line, parameters, in fact, reinforcement lasts as long as in Fig. 7. plotting enzymes concentration). D. Diffusion Rate Furthermore, Fig. 7 and Fig. 8 show, respectively, how concentration and stoichiometry can influence MoK reinforce- As regards MoK diffusion reaction, the topology depicted ment reaction behaviour, effectively modeling situatedness— in Fig. 9 has been taken as a reference. Namely, four MoK in particular, what we called the “type” of (inter-)actions. In compartments are imagined to be connected one to each other, fact, (i) in Fig. 7 the initial concentration of “red” enzymes allowing in principle any molecule to move anywhere. Fig. 8. Atoms stoichiometry increment effect on reinforcement. Fig. 11. MoK diffusion reaction trend. The yellow line plots the concentra- tion level of the atoms in their “origin” compartment (the orange horizontal line represents their source). they are functionally related. As a side note, notice a diffusion reaction featuring the law of mass action is not depicted. The motivation is that it exhibits an unexpected “malfunctioning” affecting also other reactions. More on this “interference problem” in next section. E. On the Problem of Interference Between Reactions All the experiments in the paper have been conducted incrementally, that is, each MoK reaction has been added to the BioPEPA specification one at a time. As reported in [15], when adding diffusion to other MoK behaviours, BioPEPA plots highlighted some interference between reactions. E.g., Fig. 12 depicts what happened when reinforcement has been Fig. 9. MoK topology to experiment with diffusion reaction. added to injection, decay and diffusion. Our main desiderata regarding MoK diffusion reaction are similar to those of MoK injection reaction: on one hand, we would like to perpetually spread information around, because agents working in other compartments may be interested in it; on the other hand, we would also like to keep some degree of control about “how much” information is moved around. Such “degree of control” can be achieved by reusing the concept of “saturation”, as shown by Fig. 11: in particular, it seems reasonable to allow only a fraction of molecules to move from their “origin” compartment—see Fig. 10. In practice, we can 1 // diffusion weight 2 DW = 0.75; 3 // diffusion functional rates (a@x => a@y) 4 diffSE = [DW * as@sports - as@economics]; // blue line 5 diffSC = [DW/2 * as@sports - as@crime]; // red line 6 diffSP = [DW/3 * as@sports - as@politics]; // green line Fig. 10. Notation r@c refers to the concentration of reactant r in compartment c. Previous listings did not follow such notation because there was only a Fig. 12. MoK reinforcement reaction addition to injection, decay and single compartment—MoK diffusion was not considered. diffusion. Not only enzymes are not fully depleted, but also undesirable and unexpected interferences with other reactions are clearly highlighted. arbitrarily decrease/increase the saturation-level of the origin compartment in the destination compartment. Furthermore, A number of unexpected behaviours can be seen: • first of all, our desiderata for MoK reinforcement principled and disciplined – namely, engineered – approach reaction are not met (dashed blue line). In particu- to parameter tuning we called parameter engineering. lar, it seems atoms cannot go beyond their original compartment concentration level (yellow line) IV. C ONCLUSION & F URTHER W ORKS ROADMAP • second, enzymes are not fully depleted (orange line) In this paper, we showed that simply imitating nature as • last but not least, other atoms are affected by a is may be not the optimal approach while engineering nature- successful application of MoK reinforcement reaction inspired MAS. Indeed, once a suitable natural metaphor has (yellow, red and green lines): in particular, in the time been found, MAS designers should ask themselves if the interval during which enzymes are consumed all other natural system’s parameters are the optimal ones also for the trends experiment some fluctuations artificial system they aim to build. If it is not the case, they should clearly state which goals their MAS is pursuing then The reason at the root of all these issues is still unknown: detects, preferably within the MAS itself, which parameters being chemical-like reactions scheduling essentially based on better suits their needs as well as which (if any) functional race conditions between the correspondent functional rates dependencies between such parameters better cope with the – evaluated at a given point in time –, understanding what problem their MAS aims to solve. exactly happens within the system at a given time step is not In particular, we focussed on the case of biochemical trivial at all—or even impossible, depending on the debugging coordination in MoK , showing that sticking with the law of services the simulation tool adopted provides. Nevertheless, mass action for rate expressions is not enough to model inter- the satisfactory BioPEPA specification shown in Fig. 13 has esting behaviours. Furthermore, designing more complex rate been found. In particular, MoK reinforcement reaction rate expressions demands for a principled approach going beyond has been added a “feed factor” parameter, used to weight the parameter tuning, which we call parameter engineering, likely influence of the atoms to be reinforced w.r.t. the concentration to be supported by incremental simulation of each single basic of the corresponding source in the compartment the latter “law of nature” in play. belongs to. Fig. 14 shows that our desiderata are now met To the best of our knowledge, no closely related works 1 // feed factor > 1 exists to date except, to some extent, [9]. Nevertheless, we 2 FF = 2; believe our work to be complementary to that in [23] about 3 // option (1)-revised 4 feedEC = [se@economics / (ae@crime * FF)]; self-organising design patterns as well as to [9]: in fact, once a design pattern has been recognized as a potential solution to a given problem, a simulation stage is out of doubts useful, Fig. 13. Adjusted BioPEPA specification of rate expressions for MoK therefore a parameter engineering phase necessary. reinforcement reaction used together with MoK diffusion reaction. As a last note, further works will be devoted to analyze successfully. Although not shown here for the lack of space, the tradeoff between designing more complex expressions and also the functional dependencies on enzymes concentration and sticking with the law of mass action at the cost employing more atoms stoichiometry shown in Fig. 7 and Fig. 8 are preserved. (dual, complementary and/or opposite) reactions to reach the same “complex trend” by composition. 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