-To cope with nowadays MAS complexity, natureinspired coordination models and languages gained increasing attention: in particular, biochemical coordination models. Being intrinsically stochastic and self-organising, the effectiveness of their outcome likely depends on a correct parameter tuning stage. In this paper, we focus on chemical reactions rates, showing that simply imitating chemistry “as it is” may be not enough for the purpose of effectively engineer complex, self-organising coordinated systems such as MoK .
Nowadays MAS are demanding new paradigms and abstractions to deal with their increasing complexity [1]. Such complexity is mostly due to the number and nature of interactions happening within and between MAS [2], [3]. Coordination models and languages – whose main goal is to govern such interactions [4] – have historically drawn inspiration from self-organising coordination in natural phenomena— e.g., pheromone-based [5] and chemical [6] coordination. Among the many, biochemical coordination has been shown 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 agents “implicitly” obey – as they are the “laws of nature” – drive MAS coordination. Thus, it happens by emergence as a consequence of the self-organisation between MAS coordinated entities. Being a self-organising process, coordination effectiveness likely depends on a correct parameter tuning stage, often performed in loop with a simulation stage [9].
In the case of biochemical coordination, being the “laws of nature” the (artificial) chemical reactions installed in the coordination medium, such parameters are, e.g., the rate of application of a chemical reaction, the concentration of the chemicals participating the reaction and their stoichiometry— the “extent” to which chemicals participate. In this paper, we focus on rates, aiming at a twofold goal: on one hand, showing that the law of mass action for rate expressions may be not enough to effectively engineer a biochemical coordination middleware on the other hand, highlighting that designing arbitrary functional rates demands for a disciplined and principled approach different from “parameter tuning”, which we call parameter engineering.
In particular, such goals are defined w.r.t. the MoK model and the BioPEPA tool, used as the subject and the means of investigation, respectively. Nevertheless, a generalisation of such goals suitable for any kind of nature-inspired MAS is possible. First of all, the fact that a given natural system works properly relying on a given set of parameters, each of which has a given set of functional dependencies with others, doesn’t necessarily mean that the same sets of parameters and functional dependencies will work for an artificial system drawing inspiration from the natural one. Then, to proficiently identify the relevant parameters and engineer their (possibly reciprocal) functional dependencies, a proper methodology is needed—which will likely rely on simulations.
Accordingly, the remainder of the paper is organized as follows: Section II explains what biochemical coordination is (Subsection II-A) and reminds the importance of simulation tools for self-organising systems development (Subsection II-B), also describing the MoK model (Subsection II-A1) as well as the BioPEPA tool (Subsection II-B1); Section III conveys the main contribution of the paper, introducing and motivating the notion of parameter engineering through a number of functional rates engineering examples; finally, Section IV concludes also giving some hints about further works.
II.
The chemical metaphor appears particularly appealing for MAS coordination due to the simplicity of its foundation [6]. The idea is to coordinate any MAS entity (agents as well as information) as “molecules” floating in a chemical “solution”, whose evolution is driven by chemical “reactions” continuously and spontaneously consuming and producing molecules. As many chemical reactions can occur at a given time, chemical solution evolution is driven by race conditions among their rates, which means certain reactions are stochastically executing over others—as in chemistry actually is [10].
Biochemical tuple spaces [7] enhance such metaphor by adding a spatial abstraction: the compartment. A compartment is a tuple space equipped with biochemical reactions, driving the evolution of the molecules floating in it. Compartments may be networked in “neighbourhoods” as in chemistry happens through membranes, so as to shape more complex spatial structures—such as tissues and organs. Computationally, biochemical tuple spaces are a stochastic extension of the LINDA model [11]: the idea is to equip each tuple with a “concentration” value, representing a measure of the pertinency/activity of the tuple (molecule) within the space (compartment)— the higher it is, the more likely and frequently the tuple will influence system coordination [7]. Such concentration is evolved by biochemical rules installed into the compartment, affecting concentration values over time exactly in the same way chemical substances evolve into chemical solutions [10]— that is, according to the law of mass action [12], [10].
The law of mass action is1 a mathematical model that explains and predicts the behaviour of solutions in dynamic equilibrium. It can be described with respect to two aspects: (i) the equilibrium aspect, concerning the composition of a reaction mixture at equilibrium and (ii) the kinetic aspect, concerning the rate equations for elementary reactions. The law states that the rate of an elementary reaction (rf ) – a reaction that proceeds through only one transition state, that is one mechanistic step – is proportional (kf ) to the product of the concentrations of the participating molecules (R1; R2): rf = kf [R1][R2] kf is called rate constant and, in chemistry, is a function of participating molecules affinity—to learn more, please refer to [12] and therein cited bibliography.
The MoK model, briefly described in next section, models MAS coordinated entities as well as coordination processes by (i) adopting the chemical metaphor abstractions and (ii) borrowing (to some extent) from biochemical tuple spaces the computational model.
1) The MoK Model: M olecules of K nowledge [13] (MoK for short) is a model for knowledge self-organisation in MAS. The main goals of MoK are: to let information chunks autonomously aggregate into heaps of knowledge to let knowledge autonomously flow toward the interested agents—rather than be searched Here follows a brief summary of MoK model components— consider reading [13] and [14] for MoK formalisation and early application respectively:
MoK atoms — produced by a given source to convey an “atomic piece of information”, atoms should also store some metadata to ease semantic characterisation MoK molecules — “heaps” for information aggregation, they cluster together semantically related atoms MoK enzymes — enzymes reify knowledge-oriented (inter-)actions made by agents and are meant to influence molecules’ concentration2 FM oK function — as a knowledge-driven model, MoK must have a way to determine the semantic correlation between information, therefore, the MoK function FM oK should be defined, taking two molecules and returning a value m 2 [0; 1].
MoK reactions — the behaviour of a MoK system is determined by biochemical reactions, which stochastically – according to their rate – drive molecules aggregation, reinforcement, decay, and diffusion: Aggregation3 — bounds together semantically related molecules Reinforcement — consumes an enzyme to reinforce the related molecule enzyme(Molecule1) + Moleculec1 7 !rreinf
Moleculec1+1
Moleculec 7 !rdecay Moleculec 1
fMolefcMuloel1ecSulMeso1lgeciul+esf1Mgoil+ecuflMeso2leScuMleosl2egcuilie17g!iridi usion These four biochemical reactions are the minimum set of coordination mechanisms believed (at the moment) to be necessary and sufficient to properly drive a MoK -coordinated MAS toward the desired behaviour regarding knowledge selforganisation. Nevertheless, this set may be refined and extended if its lack of expressiveness w.r.t. MoK desiderata becomes evident. Anyway, having a well-defined set of primitives is a necessary step to start distinguishing what sorts of selforganising behaviours can and cannot be achieved with MoK .
In fact, once such primitives are fixed, we can focus on the issue of properly engineering their rate expressions—rreinf , rdecay, rdiffusion. In particular: is it sufficient to stick with the law of mass action to achieve MoK goals, or should we build our “custom” functional dependencies? If so, which parameters and which kind of dependencies (direct, inverse, etc.) are to be used in each rate expression? And how can MAS designers make such decisions?
Section III answers these questions through a number of examples exploiting the BioPEPA simulation tool – briefly described in next section – to analyse different alternatives regarding MoK reactions rate expressions.
Simulation has been widely recognized as a fundamental development stage in the process of designing and implementing both MAS as well as biochemical processes [16], [9]. This is mostly due to the high number of system parameters needed, the huge number of local interactions between components, the influence of randomness and probability on system evolution. A number of different simulation tools capable of modeling biochemical-like processes exist, either born in the biochemistry field (see [17] for a survey) or in the (Multi-)Agent Based Simulation research area (survey in [18]). Among the many, ALCHEMIST [19], PRISM [20], and BioPEPA [12] at least, are worth to be mentioned. Our choice fell on the latter for its appealing features – briefly described in next section – which perfectly suit the purpose of the paper.
2Please, notice that an atom is a “singleton molecule”, hence the term “molecule” will be used also for “atom” from now on.
3Aggregation reaction formalisation is not shown here because it has been left out from BioPEPA simulations for the lack of expressiveness of the tool. To learn more, please refer to the technical report [15].
1) The Bio-PEPA Tool: BioPEPA [12] is a language for modeling and analysis of biochemical processes. It is based on PEPA [21], a process algebra originally aimed at performance analysis of software systems, extending it to deal with some features of biochemical networks, such as stoichiometry and different kinds of kinetic laws—including the law of mass action. The most appealing features of BioPEPA are: custom kinetic laws represented by means of functional rates definition of stoichiometry (“how many” molecules of a given kind participate) and role played by the species (reactant, product, enzyme, . . . ) in a given reaction theoretical roots in CTMC semantics—behind any BioPEPA specification lies a stochastic labelled transition system modeling a CTMC In BioPEPA, rate expressions are defined as mathematical equations involving reactants’ concentrations (denoted with the reactant name and dynamically computed at run-time) and supporting mathematical operators (e.g. exp and log functions) as well as built-in kinetic laws (e.g. the law of mass action, denoted with the keyword fMA) and time dependency (through the variable time, changing value dynamically according to the current simulation time step)4. The BioPEPA Eclipse plugin5 is the tool used in next section to investigate MoK reactions’ rates influence on system behaviour.
III.
As far as nature-inspired MAS are concerned, simulation tools are usually exploited to study how those parameters inherited from the natural metaphor influence the overall MAS behaviour. This is done with the aim to fine-tune such parameters value so as to get the better run-time “performances”— whatever this means (often, a behaviour closer to that exhibited in nature).
But, what about the question of wether the natural system’s parameters are well suited also for the artificial one? In particular, w.r.t. biochemical coordination (thus, MoK also): what about shaping our own rate expressions for biochemical reactions rather than blindly relying on the law of mass action to define their functional dependencies? Do we gain any improvement w.r.t. the overall coordinated MAS behaviour? Furthermore: can the same improvement be achieved by simply fine-tuning the natural system’s parameters as they are in nature (e.g. the law of mass action constant rate)?
Through the following experiments, we aim at answering this kind of questions, hopefully achieving our twofold goal: showing that the law of mass action is too weak to effectively express a number of self-organising behaviours—such as MoK ’s highlighting that shaping custom functional dependencies for rate expressions is a complex task demanding
5Instructions on how to install at http://homepages.inf.ed.ac.uk/s9552712/ bio-pepa/download.html, manual at http://homepages.inf.ed.ac.uk/stg/research/ biopepa/eclipse/manual/manual.pdf a well-engineered approach—indeed, parameter engineering prior to parameter tuning By generalisation, our first goal aims at showing there is the need to consider re-engineering natural system’s parameters, as well as their functional dependencies, so as to better cope with the problem at hand—as done with other natural metaphors: most notably, the ACO approach to distributed optimisation, in which the original “ant” metaphor is indeed just a metaphor, not the actual implementation [22].
Therefore, for each of the following experiments, we (i) identify which are the desiderata for the MAS run-time behaviour, (ii) engineer rates by designing functional dependencies which are likely to pursue the chosen goal, (iii) include a pure parameter tuning stage to fine-tune the MAS behaviour (if needed). All of this is done one reaction (coordination policy) at a time, thus one functional rate at a time, incrementally accumulated until composing the whole MAS behaviour. This approach is what we call parameter engineering.
Furthermore, a principle we believe to be extremely important for engineering self-organsing systems will be kept in mind: keeping the number of external parameters as small as possible. For “external” we mean parameters which are ad hoc added to the coordination model – MoK in our case – to better design functional rate expressions—e.g., the law of mass action constant rate. On the contrary, internal parameters are those already present in the coordinated MAS—e.g., in the case of MoK , the concentration of the reactants or the time flowing. The advantage of using internal parameters as opposed to external ones, lies in the fact that a system using more internal parameters than external ones is much more adaptive and selfregulating, since it only relies on “within-system” information rather than on “outside-system” data to dynamically adjust its behaviour—in the case of MoK , reactions’ rates.
experiments has been performed by using Gillespie’s stochastic simulation algorithm in 30 independent replications. Each of the following plots has been directly generated from BioPEPA as a result of the correspondent experiment—hence, of the 30 Gillespie runs. In each chart, the x-axis plots the time steps of the simulation, whereas the y-axis the concentration level of the reactants expressed in units of molecules.
Althought injection of atoms into a MoK compartment is not yet part of MoK ’s core set of formalised reactions, its influence on the system is so important to deserve its own analysis. Basically, injection can be described as follows:
source(Molecule 1) + Molecule c1 7 !rinj
source(Molecule 1) + Molecule c1+1 Two contrasting needs have to be addressed: on one hand, atoms should be perpetually injected into the MAS, since there is no way to know a-priori when some information will be useful; on the other hand, we would likely avoid flooding the system without any control on how many atoms are in play. Thus, three options are viable:
1 // option (
Clearly, using rate expressions based on the law of mass
action is out of question: its behaviour follows none of MoK
injection reaction desiderata. Once discarded also option (
MoK decay reaction is an effective way to resemble the relationship between information relevance and time flow. Furthermore, decay enforces a kind of negative feedback which, together with the positive feedback provided by MoK enzymes, 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
dependency (blue dashed line)
2) logarithmic time dependency + relative concentration
dependency (red line)
3) linear time dependency + built-in law of mass action
(green dashed line)
Fig. 4 shows the BioPEPA functional rates specification used6.
Again, the law of mass action is unsatisfactory, as well as
option (
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
w.r.t. a piece of knowledge – an atom or a molecule –
(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
agents (inter-)actions along a number of dimensions: time,
space, type—a “search” action, a “read” action, etc. For these
reasons, MoK reinforcement reaction rate should:
be prompt, that is rapidly increase
concentration—despite decay
molecules
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
with the law of mass action alone is out of question: option
(
Furthermore, Fig. 7 and Fig. 8 show, respectively, how concentration and stoichiometry can influence MoK reinforcement reaction behaviour, effectively modeling situatedness— in particular, what we called the “type” of (inter-)actions. In fact, (i) in Fig. 7 the initial concentration of “red” enzymes (red line) is doubled w.r.t. “yellow” enzymes (yellow line) in Fig. 6: as a result, the “duration” of the feedback is doubled as well; (ii) in Fig. 8 the stoichiometry of “red” atoms (red line) in reinforcement reaction is doubled w.r.t. “yellow” atoms (yellow line) in Fig. 6: as a result, the “intensity” of the feedback is more than doubled.
Notice also: (i) Fig. 7 shows the opposite holds too, that is, halving the initial concentration halves the duration of the feedback (yellow and blue lines); (ii) Fig. 8 shows that no interference happens between concentration and stoichiometry parameters, in fact, reinforcement lasts as long as in Fig. 7.
As regards MoK diffusion reaction, the topology depicted in Fig. 9 has been taken as a reference. Namely, four MoK compartments are imagined to be connected one to each other, allowing in principle any molecule to move anywhere. Fig. 8. Atoms stoichiometry increment effect on reinforcement.
MoK topology to experiment with diffusion reaction.
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 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 single compartment—MoK diffusion was not considered. arbitrarily decrease/increase the saturation-level of the origin compartment in the destination compartment. Furthermore, 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.
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 added to injection, decay and diffusion.
A number of unexpected behaviours can be seen:
first of all, our desiderata for MoK reinforcement
reaction are not met (dashed blue line). In
particular, it seems atoms cannot go beyond their original
compartment concentration level (yellow line)
second, enzymes are not fully depleted (orange line)
last but not least, other atoms are affected by a
successful application of MoK reinforcement reaction
(yellow, red and green lines): in particular, in the time
interval during which enzymes are consumed all other
trends experiment some fluctuations
The reason at the root of all these issues is still unknown:
being chemical-like reactions scheduling essentially based on
race conditions between the correspondent functional rates
– evaluated at a given point in time –, understanding what
exactly happens within the system at a given time step is not
trivial at all—or even impossible, depending on the debugging
services the simulation tool adopted provides. Nevertheless,
the satisfactory BioPEPA specification shown in Fig. 13 has
been found. In particular, MoK reinforcement reaction rate
has been added a “feed factor” parameter, used to weight the
influence of the atoms to be reinforced w.r.t. the concentration
of the corresponding source in the compartment the latter
belongs to. Fig. 14 shows that our desiderata are now met
1 // feed factor > 1
2 FF = 2;
3 // option (
Fig. 13. Adjusted BioPEPA specification of rate expressions for MoK reinforcement reaction used together with MoK diffusion reaction. successfully. Although not shown here for the lack of space, also the functional dependencies on enzymes concentration and atoms stoichiometry shown in Fig. 7 and Fig. 8 are preserved.
This clearly demonstrates the intricacies behind rates design in biochemical coordination, therefore motivating the principled and disciplined – namely, engineered – approach to parameter tuning we called parameter engineering.
IV.
In this paper, we showed that simply imitating nature as is may be not the optimal approach while engineering natureinspired MAS. Indeed, once a suitable natural metaphor has been found, MAS designers should ask themselves if the natural system’s parameters are the optimal ones also for the artificial system they aim to build. If it is not the case, they should clearly state which goals their MAS is pursuing then detects, preferably within the MAS itself, which parameters better suits their needs as well as which (if any) functional dependencies between such parameters better cope with the problem their MAS aims to solve.
In particular, we focussed on the case of biochemical coordination in MoK , showing that sticking with the law of mass action for rate expressions is not enough to model interesting behaviours. Furthermore, designing more complex rate expressions demands for a principled approach going beyond parameter tuning, which we call parameter engineering, likely to be supported by incremental simulation of each single basic “law of nature” in play.
To the best of our knowledge, no closely related works exists to date except, to some extent, [9]. Nevertheless, we believe our work to be complementary to that in [23] about 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, therefore a parameter engineering phase necessary.
As a last note, further works will be devoted to analyze the tradeoff between designing more complex expressions and sticking with the law of mass action at the cost employing more (dual, complementary and/or opposite) reactions to reach the same “complex trend” by composition.
This work has been supported by the EU-FP7-FET Proactive project SAPERE – Self-aware Pervasive Service Ecosystems, under contract no. 256874.