Weak Bisimulation Between TwoBiogeochemical Cycles

Jhoirene B. Clemente1, John Justine S. Villar2, and Henry N. Adorna1

1 Algorithms and Complexity Laboratory, Department of Computer Science,University of the Philippines Diliman, Quezon City, Philippines

jbclemente@up.edu.ph, hnadorna@dcs.upd.edu.ph2 Scientific Computing Laboratory, Department of Computer Science,

University of the Philippines Diliman, Quezon City, Philippinesjohn justine.villar@up.edu.ph

Abstract. In this paper, we show that the two biogeochemical pro-cesses, carbon and nitrogen cycles are weakly bisimilar or weakly equiv-alent. We model each process as labeled transition system (LTS), wheretransitions between Carbon and Nitrogen states are movements fromdifferent compartment of Earth. We showed that there exist a binaryrelation S over the states of the two systems, such that one weakly sim-ulates the other with respect to their changes in location.

1 Introduction

Bisimulation, a binary relation between two communicating systems, is con-sidered as one of the most important concepts in concurrency theory, whichstudies a family of simultaneously evolving components that interacts (i.e., com-municates) synchronously. Roughly defined, a bisimulation is a binary relationbetween labeled transition systems (LTS), with an associated system which be-have (i.e., simulates) the other system [4]. Moreover, the bisimulation equality,called bisimilarity, is the most studied form of behavioral equality for processesas it is used to abstract from certain details of the systems of interest [4].

Bisimulation is widely applied to various fields of computer science, but notlimited to, functional and object-oriented languages, data structures, domains,databases, compiler optimization, program analysis, verification, and others. Inthis paper, the authors attempt to apply the concepts in concurrency theory totwo biogeochemical processes to provide preliminary analyses, as well as predic-tions on the biological systems of interest.

A biogeochemical cycle is defined as a pathway by which a chemical elementor molecule moves through both biotic (biosphere) and abiotic (lithosphere, at-mosphere, and hydrosphere) compartments of Earth. This interests the scientistsas these cycles significantly affect the biotic components (plants and animals) asthey draw carbon dioxide and nitrogen (in plants) to perform bodily processes.The exchange of chemical nutrients impact some of the Earth’s compartmentsas some of these release more of the nutrient or absorbed by a compartment

faster, which may harm several interacting components involved in the cycle.Most studies intend to show how changing climate alter the carbon and nitrogencycles, and vice-versa. Another interesting fact is that the starting state of thecycle is the gaseous state of the elements (carbon and nitrogen), and is absorbedby the plants and animals as a first chemical process. This may provide somelead on how different forms of carbon and nitrogen transfer between Earth’scompartment.

The main goal of this paper is to draw several theoretical hypotheses aboutthe behavior of the carbon and nitrogen cycles, given the location of the chemicalforms of carbon and nitrogen.

2 Carbon Cycle

Carbon, the fourth most abundant element in the Universe, is both the foun-dation of all life on Earth and the source of the majority of energy consumedby human civilization [7]. Most of Earth’s carbon – about 65,500 billion metrictons – is stored in rocks, and the rest is in the ocean, atmosphere, plants, soil,and fossil fuels.

Its flow, called the carbon cycle [5], between each Earth’s reservoirs hasbeen heavily studied by ecologists, climatologists and marine scientists, amongothers, as it directly affects global temperature, climate ecological change. Theimbalance of carbon between the biotic and abiotic compartments primarilyharm the different living things because excess carbon in the atmosphere warmsthe planet and helps plants on land grow more. Moreover, excess carbon in theocean makes the water more acidic, putting marine life in danger.

Note that carbon takes different chemical forms all through out the carboncycle, in which the important forms considered in this paper are the following:

– pure carbon (C) – present commonly in the Earth’s lithosphere– carbon monoxide (CO) – present in the Earth’s atmosphere– carbon dioxide (CO2) – present in the Earth’s atmosphere– methane (CH4) – present in the Earth’s atmosphere– glucose ((CH2O)6) – present in the Earth’s biosphere

Also, the processes involved in the carbon cycle are summarized below:

– Photosynthesis– Respiration– Decomposition– Volcanic activity– Erosion– Deposition– Diffusion– Industrial Activities

Furthermore, Figure 1 illustrates the flow of carbon in the different biogeo-chemical spheres.

Fig. 1: The Carbon Cycle

3 Nitrogen Cycle

Nitrogen is a colorless, odorless, tasteless, and mostly inert gas at standard con-ditions. It constitutes 78.09% by volume of the Earth’s atmosphere. Nitrogenis a required nutrient for all living organisms to produce a number of complexorganic molecules like amino acids, the building blocks of proteins, and nucleicacids, including DNA and RNA. In ecology, the abundance of nitrogen is mea-sured to identify the amount of food that can be grown in a piece of land [1].

Nitrogen can take different forms all through out the nitrogen cycle. Asidefrom the organic nitrogen (N) which is present inside living organisms, nitrogencan take the following molecular forms, alongside with the location where theyare present.

– Inorganic nitrogen (N2) - present in the Earth’s atmosphere– Ammonium (NH+

4 ) - present in the Earth’s lithosphere– Ammonia (NH3) - present in the Earth’s atmosphere– Nitrite (NO−2 ) - present in the Earth’s lithosphere– Nitrate (NO−3 )- present in the Earth’s lithosphere– Nitrous oxide (N2O) - present in the Earth’s atmosphere– Nitric oxide (NO) - present in the Earth’s atmosphere

Inorganic nitrogen (N2) is abundant in the Earth’s atmosphere. However,it is unusable in its current form. The inorganic nitrogen undergoes nitrogenfixation through lightning and nitrogen fixing bacteria to become usable. In the

Fig. 2: Nitrogen Cycle

atmosphere, lightning converts N2 to NO, and when it precipitates the NOin the atmosphere reach the ground as nitrates NO3. Nitrogen fixing bacteriathat are present in roots of leguminous plants absorbs N2 in the atmosphereand releases ammonium (NH4), which is then converted to ammonia (NH3).Industrial processes can also convert N2 to either nitrate or nitrite throughcombustion.

Nitrogen from the soil is absorbed by plants through the process of assimila-tion. The nitrogen from living organisms such as plants and animals are releasedto the Earth’s lithosphere during decomposition and egestion, nitrogen moleculesare converted into ammonium and ammonia. The process is called ammonifica-tion. The complete illustration of processes involved in Nitrogen cycle is shownin Figure 2. A detailed list of transition from one nitrogen state to another ispresented in Section 4.2.

4 Labeled Transition System Construction

4.1 Preliminaries

The relevant terms used in the following sections are provided below:

Definition 1 (Labeled Transition System (LTS)). A labeled transition sys-tem (LTS) over Act is a pair (Q, T ) consisting of

– a set Q of states

– a ternary relation T ⊆ Q×Act×Q, known as the transition function.

If (q, α, q′) ∈ T we write qα→ q′, and we call q the source and q′ the target of

the transition.

Definition 2 (Strong Simulation). Let (Q, T ) be an LTS, and let S be abinary relation over Q. Then S is a strong simulation over (Q, T ) if, wheneverpSq,

if pα−→ p′ then there exists q′ ∈ Q such that q

α−→ q′ and p′Sq′.

Definition 3 (Converse of a Binary Relation). The converse R−1 of anybinary relation R is the set of pairs (y, x) such that (x, y) ∈ R.

Definition 4 (Strong Bisimulation, Strong Equivalence). A binary rela-tion S over Q is said to be a strong bisimulation over the LTS (Q, T ) if both Sand its converse are simulations. We say that p and q are strongly bisimilar orstrongly equivalent, written p ∼ q, if there exists a strong bisimulation S suchthat pSq.

Definition 5 (Experiment Relations). The relations ⇒ ands⇒, for any s ∈

Act∗, are defined as follows:

1. P ⇒ Q means that there is a sequence of zero or more reactions P → . . .→Q. Formally, ⇒def

=→∗, the transitive closure of →.2. Let s = α1 . . . αn. Then P

s⇒ Q means P ⇒α1→ P1 . . . ⇒αn→ Pn ⇒ Q. For-

mally,s⇒def

=⇒α1→⇒ . . .⇒αn→⇒.

Definition 6 (Weak Bisimulation). Let S be a binary relation over P. ThenS is said to be a weak simulation if, whenever PSQ,

if Pe⇒ P ′ then there exists Q′ ∈ P such that Q

e⇒ Q′ and P ′SQ′.

We say that Q weakly simulates P if there exists a weak simulation S such thatPSQ.

4.2 The Nitrogen and Carbon Cycle LTSs

Assumptions. In this paper, we assume the following

– We only considered presence of carbon and nitrogen (with its chemical by-products) in the atmosphere, lithosphere, hydrosphere and biosphere,as well as man-made industrial influences.

– We only considered major chemical compounds that mainly participate inthe carbon and nitrogen cycles.

– We also assume that other chemical compounds (without carbon/nitrogen)involved in the reaction (LHS and RHS of the equation) are disregardedin the construction of the LTS. For example, in the chemical process ofphotosynthesis, given by CO2 +H2O + (Nutrients) → (CH2O)6 +O2, weonly considered CO2 and (CH2O)6 to participate in the transition.

Carbon Cycle LTS. The LTS for carbon cycle is (Qc, Tc), where the set Qcincludes the different forms of carbon, i.e. Qc = {C, CO, CO2, CH4, (CH2O)6}and Tc = Q× x ×Q with x ∈ Actc ⊆ Lc × Lc, Lc = {a, l, h, b, i}. Here a, l, h, band i corresponds to the atmosphere, lithosphere, hydrosphere, biosphere, andthe industrial domain. In the construction of transitions, x is stated without theparentheses (instead of the ordered pair notation) for simplicity, i.e., x is writtenas yz instead of (y, z).

The transitions of the LTS are derived from the chemical processes involvedin conversion of one form carbon to the other. Since we are interested in thelocation of the chemical nutrient as it changes its form, the label of the actions arerepresented as the change of locations. The following enumerates the transitionsof the LTS involved in each process of the carbon cycle.

– photosynthesis:

CO2ab−→ (CH2O)6

– respiration, decomposition:

(CH2O)6ba−→ CO2

– photosynthesis:

CO2hb−→ (CH2O)6

– respiration, decomposition:

(CH2O)6ba−→ CH4

– decomposition: (CH2O)6bl−→ C

– decomposition: (CH2O)6bl−→ CO2

– disintegration: CO2ll−→ C

– volcanic activity: Cla−→ CO

– volcanic activity: Cla−→ CO2

– combustion: Cla−→ CH4

– combustion: (CH2O)6ba−→ CH4

– erosion: Clh−→ C

– deposition: Chl−→ C

– diffusion: CO2ah−→ CO2

– diffusion: CO2ha−→ CO2

– diffusion: CO2la−→ CO2

– industrial emission: Cia−→ CO

– industrial emission: Cia−→ CO2

– industrial emission: Cia−→ CH4

In total, we have 19 transitions. Using the set of states and the list of tran-sitions derived from the processes in the carbon cycle, as stated above, Figure 3shows the complete labeled transition system diagram.

Nitrogen Cycle LTS. The LTS for nitrogen cycle is (Qn, Tn), where the setQn includes the different forms of nitrogen, i.e.

Qn = {N,N2, NO,N2O,NOx, NO−2 , NO

−3 , NH3, NH

+4 }

and Tn = Q×x×Q with x ∈ Actn ⊆ Ln×Ln, Ln = {a, l, b}. Note that, Ln ⊂ Lc,because several locations such as h, b, and i ∈ Lc are not considered in thenitrogen cycle. Biological processes for three different location, i.e. atmosphere,biosphere, and lithosphere were only considered in the LTS. The transitions ofthe LTS are derived from the chemical processes involved in conversion of onenitrogen form to the other.

CO2

(CH2O) 6 C

CH4 CO

la, ia la, ia ba

la, ha, ah

bl

la, ia

ll

lh, hl

ba, bh, bl

ab, hb

Fig. 3: Carbon Cycle LTS

The transition labels of the actions follows our convention presented in theprevious subsection. The following enumerates the transitions of the LTS in-volved in each process of the nitrogen cycle.

– Nitrogen Fixation• lightning: N2

aa−→ NO

• precipitation: NOal−→ NO−3

• nitrogen fixing bacteria:

N2ab−→ N

• nitrogen fixing bacteria:

Nba−→ NH3

• combustion: N2aa−→ NH3

• combustion: N2aa−→ NOx

– Assimilation• NH3

ab−→ N

• NO3lb−→ N

• NH+4

lb−→ N

– Ammonification

• N bl−→ NH+4

• NH+4

la−→ NH3

– Nitrification

• NH+4

ll−→ NO−2

• NO−2ll−→ NO−3

– Denitrification

• NO−3la−→ N2

– Ammonium Oxidation

• NO−2la−→ N2

• NH+4

la−→ N2

Using the set of states we previously defined and the list of transitions de-rived from the processes of nitrogen cycle. Figure 4 shows the complete labeledtransition system diagram.

LTS with Merged State and Transition Spaces. In order to use the defi-nition of bisimulations presented in Section 2, we define a new LTS (Q, T ), withQ = Qc ∪ Qn and T = Tc ∪ Tn. The relation S will be defined over the set Q.Since we would like to assess the equivalence of nitrogen and carbon cycle, wewill not consider a pair of molecules obtained from the same subsystem.

Fig. 4: Nitrogen Cycle LTS

Fig. 5: Carbon Cycle and Nitrogen Cycle LTS with location as states

5 Reduction of Derived LTS

Note that we have two different set of actions Actc and Actn involved in bothLTS. This is primarily because molecules of carbon and nitrogen thrive in dif-ferent locations all through out the cycle. For instance, majority of carbonmolecules are produced in hydrosphere while usable nitrogen molecules are pro-duced mainly in the atmosphere and biosphere. In order to compare two differentsystems, we obtained a reduced versions of each LTS such that we can ignore sev-eral transitions involving actions that are not common to both. We can look atthe reduced LTSs as restricted versions of the two LTSs with respect to commonlocations of nitrogen and carbon molecules.

In order to do so, we derived the dual of the LTS as shown in Figure 5.The states in the corresponding LTS are the different locations of carbon andnitrogen molecules. The transition labels between two locations are represented

by an ordered pair of molecules where there exists a process that converts oneto the another in the corresponding biogeochemical process. It is shown fromFigure 5 that not all states in the LTS is present to the other. From this twoLTS, we obtained a reduced LTS where they contain the same transitions. Theresulting carbon and nitrogen cycle LTS, with transitions common to both cyclesare shown in Figure 6.

Fig. 6: Reduced Carbon Cycle and Nitrogen Cycle LTS

Note that the reduction involved removing processes or transitions involvedin each of the system, instead of collapsing several processes into one to eliminateactions not common to both. This latter type of reduction is not considered inthe study because we would like to treat each transition as an independentprocess, i.e. we do not impose ordering in the processes involved. We also intendto preserve the integrity (i.e., natural occurrence) of the processes involved inboth biogeochemical processes so that the subsequent analyses will be as soundas possible.

6 Discussion

6.1 Construction of S

Now that we have the both LTS constructed in such a way that they sharethe same actions, we try to find a set S = {(p, q)|p ∈ Qc, q ∈ Qn} that theresulting LTS derived from S gives us a bisimulation. The list shown in Table1 provides a transition (p, q)→ (p′, q′) which means that the transitions p→ p′

and q → q′ have the same actions. Equations (1)-(2) have the actions ll, (3)-(18)have the actions la, (19)-(20) have the actions ab, (21)-(22) have the actions ba,and (23)-(25) have the actions bl.

In order to obtain the set S, we construct a transition graph G, where wecan start on any arbitrary node (p, q). A directed edge e from (p, q) to (p′, q′)exists, if transition is found in Table 1. Based from Definition 6, we can obtainS by getting a set of vertices V (GS) ⊆ V (G) such that for all vertices v ∈ V (GS),

(p,q) → (p′,q′)

(CO2, NO2) → (C,NO3) (1)

(CO2, NH4) → (C,NO2) (2)

(C,NH4) → (CO,N2) (3)

(C,NO2) → (CO,N2) (4)

(C,NH4) → (CO,NH3) (5)

(C,NO3) → (CO,N2) (6)

(C,NH4) → (CO2, N2) (7)

(C,NO2) → (CO2, N2) (8)

(C,NH4) → (CO2, NH3) (9)

(C,NO3) → (CO2, N2) (10)

(C,NH4) → (CH4, N2) (11)

(C,NO2) → (CH4, N2) (12)

(C,NH4) → (CH4, NH3) (13)

(p,q) → (p′,q′)

(C,NO3) → (CH4, N2) (14)

(CO2, NH4) → (CO2, N2) (15)

(CO2, NO2) → (CO2, N2) (16)

(CO2, NH4) → (CO2, NH3) (17)

(CO2, NO3) → (CO2, N2) (18)

(CO2, N2) → ((CH2O)6, N) (19)

(CO2, NO3) → ((CH2O)6, N) (20)

(CO2, NH3) → ((CH2O)6, N) (21)

((CH2O)6, N) → (CO2, NH3) (22)

((CH2O)6, N) → (CH4, NH3) (23)

((CH2O)6, N) → (C,NH4) (24)

((CH2O)6, N) → (CO2, NH4) (25)

Table 1: List of all transition pairs derived from the reduced Nitrogen and Carboncycle LTSs with the same action labels.

there exists an edge between v and w where w ∈ V (GS). The vertices in V (GS)are the pair of states in S.

Fig. 7: Transition Graph G obtained from the list of transitions in Table 1. Thepair included in set S are the nodes with darker shade.

From the transition graph G in Figure 7, we can infer that S = {(CO2, NO2),(C,NO3), (CO2, N2), ((CH2O)6, N), (CO2, NH3), (CO2, NH4), (C,NO2)}, which

has 7 ordered pairs. The ordered pairs in a lighter shade are not included in Sas they does not induce a cycle. Note also that the entries of the ordered pair(p, q) belong to the same compartment, i.e., p, q ∈ L.

6.2 Further Analysis

We constructed two pairs of LTS in which the states come from the entries ofthe ordered pairs in S. Note that in S, we have two ordered pairs which havethe same second entry, i.e. (CO2, NO2) and (C,NO2), which results to twocomparisons. Figures 8 (a) and (b) shows the resulting LTS derived from S forthe carbon and nitrogen cycles, respectively.

Fig. 8: Derived Carbon Cycle (a) and Nitrogen (b) LTS

In the two cases, we considered the ordered pairs (CO2, NO2) and (C,NO2)respectively as starting points. Both cases in Figure 9, infer that LTS fromNitrogen cycle and Carbon cycle are bisimilar.

Fig. 9: (a) Case 1: Used (CO2, NO2) as a starting point (b) Case 2: Used(C,NO2) as a starting point

7 Conclusion and Future Works

In this paper, we have shown a weak bisimulation between Carbon and Nitrogencycle. We constructed a general and restricted versions of the LTS to be able toprovide a valid comparison. In showing the bisimulation between the two sys-tems, we defined a larger LTS, (Q, T ) containing both LTS from Carbon and Ni-trogen cycle. We then defined a relation S = {(CO2, NO2), (C,NO3), (CO2, N2),((CH2O)6, N), (CO2, NH3), (CO2, NH4), (C,NO2)} obtained from the set ofstates in Q. The pair of molecules from carbon and nitrogen respectively mayhelp in inferring relationships between molecules involved in analyzing complexsystems in environmental science. One possible extension of this study is to incor-porate environmental data in transition systems in order to measure quantitativeeffects of perturbation in the environment.

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