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The authors have declared that no competing interests exist.

Current address: Molecular Systems Biology, Groningen Biomolecular Sciences and Biotechnology Institute, University of Groningen, Groningen, the Netherlands

The golden ratio,

The golden ratio

The basis of Dan Brown’s fictional discussion of the golden ratio

These claims are, however, mostly false or misleading [

One area of biology in which the golden ratio has a genuine role is in phyllotaxis, i.e., the arrangement of leaves on the stem of a plant. The seeds in the head of a ripe sunflower fit neatly together into two interlocking families of helical spirals. The number of clockwise and anti-clockwise winding spirals are often both successive numbers of the Fibonacci sequence [

The limit of the ratio between two successive Fibonacci numbers is the golden ratio, namely, lim_{t→∞} _{t}/_{t−1} =

The Fibonacci process _{t} is the number of adult pairs at month _{t−1}. So the ratio of the number of adult pairs over the number of infant pairs goes to

Recently, the golden ratio was suggested to appear in another totally different field: the human genome. Dress _{A} + _{C} for the 24 human chromosomes can also be explained by the Fibonacci process _{A} is the frequency of codon A and likewise, _{C} is the frequency of codon C [_{C} + _{G})/(_{A} _{T}) is approximately (3 −

Another famous model that can generate

This process is equivalent to the Fibonacci rabbit model. Note that

These models do, from a mathematical viewpoint, give rise to the golden ratio, but our question is whether the golden ratio has a special role in natural self-replicating systems in general, as implied by the authors in papers [

To answer these questions we employ a framework of chemically realistic self-replicating reaction systems, introduced by us in paper [

This paper is organised as follows. Section

In an earlier paper [

By choosing some reactions that satisfy the requirements above, we can construct a chemically realistic reaction system, namely, an instantiation of the general framework. The following system is one of such instantiations,

We define the

In paper [

For every reaction, at least one type of its reactants comes from the products of other reactions;

There is at least one intermediate molecule that appears on the reactant side fewer times than that on the product side;

There are no intermediate molecules that appear on the reactant side more often than on the product side.

In this paper we introduce an alternative, narrower criterion under which certain chemically realistic self-replicating systems are analysable. We say that a self-replicating system is analysable if it satisfies criterion 1, 2, and the following criterion 3*:

On the reactant side of the whole system, each intermediate molecule species appears at most once.

Note that system

We now introduce a scheme for population dynamics for analysable self-replicating chemical reaction systems, using system _{i}(_{1}(0) = _{2}(0) = _{3}(0) = _{5}(0) = 1, i.e., one of each intermediate molecules. In addition, we assume that there is an infinite number of resource molecules _{4}(

We update molecule populations at _{i}(

Consider how we apply this updating procedure to system

Firstly, update _{i}(0) to obtain _{i}(1). For molecule _{1}(0) = 1, we let this one of _{2}(0) = 1, we let this one of _{3}(0) = 1, we let this one of _{5}(0) = 1, we let this one of _{1}(1) = 3, _{2}(1) = 2, _{3}(1) = 1 and _{5}(1) = 1.

Then, update _{i}(1) to obtain _{i}(2). _{1}(1) = 3 of molecule _{2}(1) = 2 of _{3}(1) = 1 of _{5}(1) = 1 of _{1}(2) = 5, _{2}(2) = 2, _{3}(2) = 1 and _{5}(2) = 3.

Similarly, at generation _{1}(3) = 5, _{2}(3) = 4, _{3}(3) = 3 and _{5}(3) = 5.

Continue this updating procedure. Finally we obtain a sequence of _{1}(_{2}(_{3}(_{5}(

Generation |
_{1}( |
_{2}( |
_{3}( |
_{5}( |
---|---|---|---|---|

0 | 1 | 1 | 1 | 1 |

1 | 3 | 2 | 1 | 1 |

2 | 5 | 2 | 1 | 3 |

3 | 5 | 4 | 3 | 5 |

4 | 11 | 8 | 5 | 5 |

5 | 21 | 10 | 5 | 11 |

6 | 25 | 16 | 11 | 21 |

⋮ | ⋮ |

Note that _{i}(_{i}(

We call this _{1}(_{2}(_{3}(_{5}(^{⊺} as

Therefore, we can also use

Eqs

Through either of the steps above, we obtain the sequences _{i}(

Specifically, lim_{t→∞} _{1}(_{t→∞} _{2}(_{t→∞} _{5}(_{t→∞} _{3}(

We now rigorously prove _{5}(_{1}(_{3}(_{5}(_{1}(_{2}(_{3}(_{5}(_{1}(_{1}(

Based on standard skills for solving recursive sequences [_{1}(_{1}(0) [_{1} is the largest among all roots, resulting in _{3} and λ_{4}. Therefore, we have

Note that

Now we show another example of a chemically realistic self-replicating reaction system that is characterised by

The characteristic equation of the recurrence matrix is

This equation has five roots, where four of them are the same as in ^{2} − λ − 1). We have now seen that this occurs for at least two analysable self-replicating systems.

Let us now consider the L-system (equivalently, the Fibonacci rabbit model) given in

The characteristic equation for this recurrence matrix is λ^{2} − λ − 1 = 0. So the characteristic number Λ for this system is

There is however a serious problem with the biological interpretation of system

We can, nonetheless, construct a chemically realistic self-replicating system which is analogous to

After omitting to write the resource (

We now investigate a larger range of chemically realistic self-replicating reaction systems in order to determine which numbers typically characterise their behaviour. We consider all of the analysable self-replicating systems up to where the possibly largest molecule is

The blue lines correspond to all

The most frequent characteristic number is _{i}(

Generation |
_{2}( |
_{3}( |
_{4}( |
---|---|---|---|

0 | 1 | 1 | 1 |

1 | 2 | 1 | 1 |

2 | 2 | 2 | 1 |

3 | 2 | 2 | 2 |

4 | 4 | 2 | 2 |

5 | 4 | 4 | 2 |

6 | 4 | 4 | 4 |

⋮ | ⋮ |

Equivalently, _{2}(_{3}(_{4}(^{⊺}. We call this type of system as the

System

The second most frequent characteristic number in ^{3} = 1 + 1/Λ. In general, the number ^{k} = 1 + 1/

The eigenvalues of the recurrence matrix are 1.22074…, −0.24812… ±

Note that the 1st lower golden ratio (namely,

In order to explain why certain characteristic numbers arise, we note that the _{jk}) for any analysable self-replicating chemical reaction system must satisfy the following conditions:

It is a non-negative integer square matrix. This is because any reaction involves only integer number of molecules.

All entries on the main diagonal are 0, namely _{jj} = 0. This is because no molecule can appear on both sides of a reaction (e.g.,

_{jk} and _{kj} cannot be both larger than 0, that is, _{jk} ⋅ _{kj} = 0. This is because a reaction and its reverse reaction cannot both appear in one system.

The sum of any column is either 0, 1 or 2. This is because of criterion 3* in Section

The sum of at least one column is 2. This is because of criterion 2 in Section

The sum of any row is at least 1. This is because of criterion 1 in Section

Therefore, the characteristic equation of _{3}, _{4}, ⋯, _{m} are all integers. Note that (1) this polynomial is always monic, i.e., the leading coefficient is 1; (2) the term λ^{m−1} is zero because of condition 2 of ^{m−2} is zero because of condition 3 of

For all of the chemically realistic self-replicating systems we investigated (where the largest molecule is

For any 4-intermediate-molecule system, the characteristic equation is

For any 5-intermediate-molecule system, the characteristic equation is

For any 6-intermediate-molecule system, the characteristic equation is

Here we list four properties of the characteristic number Λ for self-replicating systems:

Any Λ appeared in

The self-replicating systems that is characterised by the golden ratio

From

From our simulations, we observed that for any integer self-replicating system, some of its recurrence matrix

^{p} _{1}, λ_{2}, ⋯, λ_{m} are the

This hypothesis holds numerically for all the cases we have studied, and is a very striking result, but we have been unable to prove it rigorously. We can construct a specific type of graph that satisfies all the six conditions for

The scheme

In paper [^{−1})
_{+ij} is the rate constant for this synthesis reaction, _{i} = _{i}(^{−1})
_{−ij} is the rate constant for this decomposition reaction. The physical conditions and the properties of chemicals assumed in paper [_{+ij} or _{−ij} for all reactions in the self-replicating system are identical, thus denoted

We now investigate system _{1} + _{2} + _{3} + _{5}. There are only two parameters. We set ^{11} ^{−1}, the same as in paper [^{10}.

Solutions of _{1}(0) = _{2}(0) = _{3}(0) = _{5}(0) = 1, are shown in ^{−10} _{i} = _{i}/(_{1} + _{2} + _{3} + _{5}), to compare with _{τ→∞}(_{1}(_{2}(_{τ→∞}(_{1}(_{2}(

(a) Solutions of

During the exponential growth phase, however, the normalised population _{i} remains almost unchanged (_{i} during this phase is approximately identical to the corresponding limit value in _{1} ≈ 0.38196, _{2} = _{5} ≈ 0.23606, _{3} ≈ 0.14589, and thus _{1}/_{2} ≈ _{1} + _{2} + _{3} + _{5}) → 1. That is why, in ^{−10} _{1} + _{2} + _{3} + _{5} becomes of comparable magnitude of _{1} + _{2} + _{3} + _{5}) → 1,

Therefore, the characteristic number of the system are transient behaviours corresponding to the scenario that all other molecules are much fewer than the resource inside the system, equivalently, the resource inside the system is infinite. It is interesting to note that the Jacobian of the ODEs _{1} = _{2} = _{3} = _{5} = 0 is (

The transitory nature of characteristic numbers in growth is not the only reason to doubt their universal significance. Note that we also made a strong assumption about the rate constants above that they are all identical, namely, _{ij} =

There is yet another factor that limits the generality of the characteristic number of self-replicating systems. Take the following system

(a) Solutions of its corresponding ODEs in log-normal scale. Note that populations of molecule

The factors listed above, combined with the low number of analysable systems which give rise to

A wide range of chemically realistic self-replicating systems can be described by recurrence matrices that can be further characterised by an algebraic number, which is the largest absolute value among all of the eigenvalues of the recurrence matrix. In some cases, the characteristic number is the golden ratio

Our work suggests that there is no reason to believe that

We conclude that (1)

We suggest, instead, that a more useful approach is to develop models that characterise self-replication and investigate their general properties. We see our chemical system and approach as a useful step in this direction.

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Y. L. thanks Erasmus Mundus Action 2 programme (Lotus Scholarship), funded by the European Commission, for support of his Ph.D. study in Uppsala University, where the work on this paper was undertaken.