Counting Rings in a Molecule

Find the smallest set of smallest rings (SSSR) of a molecule

In chemistry and cheminformatics, counting rings in a molecule helps describe its structure, stability, and reactivity. Ring counts are used in predicting properties like aromaticity, classifying compounds, and generating molecular descriptors for drug design and materials science. Chemists use the smallest set of smallest rings (SSSR) as a consistent measure for the number of rings in a molecule.

Start with biphenyl, a small molecule with two benzene rings joined by a single bond:

In[1]:=

(m=Molecule["biphenyl"])//MoleculePlot

Out[1]=

Biphenyl clearly has two rings, which can be found by querying the "RingCount" molecule property:

In[2]:=

m["RingCount"]

Out[2]=

Naphthalene has two rings that are fused together by a shared bond:

In[3]:=

(m=Molecule["naphthalene"])//MoleculePlot

Out[3]=

The "RingCount" property also returns 2 for naphthalene:

In[4]:=

m["RingCount"]

Out[4]=

The "RingCount" property actually returns the number of rings in the smallest set of smallest rings (SSSR):

In[5]:=

m["SmallestSetOfSmallestRings"]

Out[5]=

{{1,10,9,4,3,2},{5,6,7,8,9,4}}

In[6]:=

MoleculePlot[m,%]

Out[6]=

Note that the SSSR does not include the outer envelope ring containing all 10 carbon atoms, because mathematically it can be formed by combining the smaller rings. To count all cycles, convert the molecule to a

Graph

and then call

FindCycle

In[7]:=

allCycles=FindCycle[MoleculeGraph[m],Infinity(*nosizelimit*),Infinity(*findthemall*)]

Out[7]=

{{49,98,87,76,65,54},{12,23,34,49,910,101},{12,23,34,45,56,67,78,89,910,101}}

Visualize all rings, not just those in the SSSR:

In[8]:=

Table[MoleculePlot[m,{cycle}],{cycle,allCycles/.UndirectedEdge[a_,b_]Bond[{a,b}]}]

Out[8]=





Why do chemists use the SSSR instead of enumerating all cycles in the graph? There is both a chemical and a practical answer. Consider a larger structure like this cycloamylose:

In[9]:=

m=

cyclooctaamylose

CHEMICAL

//MoleculePlot3D

Out[9]=

The SSSR for this molecule has eight rings with 6 atoms and twenty four rings with forty atoms:

In[10]:=

CountsBy[Length]@MoleculeValue[m,"SmallestSetOfSmallestRings"]

Out[10]=

68,4024

Enumerating all possible cycles increases the number of 40-atom rings to 256:

In[11]:=

CountsBy[Length]@FindCycle[MoleculeGraph[m],Infinity,Infinity]

Out[11]=

68,40256

By looking at only the SSSR a greater emphasis is placed on the smaller rings, which are much more relevant to the overall chemistry of this molecule. This alone makes the SSSR count a more useful topological descriptor for property prediction models. Another reason to prefer the SSSR is practical, namely that it can be computed much more efficiently. For example finding the SSSR for buckminsterfullerene is found very quickly:

In[12]:=

m=Molecule

buckminsterfullerene

CHEMICAL

//MoleculePlot3D

Out[12]=

In[13]:=

MoleculeValue[m,"RingCount"]//AbsoluteTiming

Out[13]=

{0.000404,32}

All of the five and six-membered rings are contained in the SSSR:

In[14]:=

CountsBy[Length]@MoleculeValue[m,"SmallestSetOfSmallestRings"]

Out[14]=

512,620

By contrast, the number of total cycles in buckminsterfullerene is exponentially large and not countable in a reasonable time:

In[15]:=

CountsBy[Length]@FindCycle[MoleculeGraph[m],Infinity,10^6]//AbsoluteTiming

Out[15]=

{9.08103,161,175,188,1912,2024,2149,2279,23103,24178,25257,26381,27613,28857,291305,301969,312982,324263,336209,349254,3512626,3617310,3723284,3829824,3937374,4046186,4155352,4263252,4371616,4478405,4581540,4682084,4779421,4872652,4962407,5051516,5139585,5228069,5318436,5411129,555751,562469,57881,58244,5933,605}

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Contributed by: Jason Biggs

Wolfram Language Example Repository

Counting Rings in a Molecule

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Counting Rings in a Molecule

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