Abstract
The golden ratio, ϕ = 1.61803…, has often been found in connection with biological phenomena, ranging from spirals in sunflowers to gene frequency. One example where the golden ratio often arises is in self-replication, having its mathematical origins in Fibonacci’s sequence for “rabbit reproduction”. Recently, it has been claimed that ϕ determines the ratio between the number of different nucleobases in human genome. Such empirical examples continue to give credence to the idea that the golden ratio is a universal constant, not only in mathematics but also for biology. In this paper, we employ a general framework for chemically realistic self-replicating reaction systems and investigate whether the ratio of chemical species population follows “universal constants”. We find that many self-replicating systems can be characterised by an algebraic number, which, in some cases, is the golden ratio. However, many other algebraic numbers arise from these systems, and some of them—such as and 1.22074… which is also known as the 3rd lower golden ratio—arise more frequently in self-replicating systems than the golden ratio. The “universal constants” in these systems arise as roots of a limited number of distinct characteristic equations. In addition, these “universal constants” are transient behaviours of self-replicating systems, corresponding to the scenario that the resource inside the system is infinite, which is not always the case in practice. Therefore, we argue that the golden ratio should not be considered as a special universal constant in self-replicating systems, and that the ratios between different chemical species only go to certain numbers under some idealised scenarios.
Highlights
IntroductionIntroduction pffiffiThe golden ratio ð1 þ 5Þ=2 1⁄4 1:61803 . . ., usually denoted by the Greek letter φ (PHI), has attracted broad attention for a long time, with suggestions that they are aesthetically significant and ubiquitous in nature
Introduction pffiffi The golden ratio ð1 þ 5Þ=2 1⁄4 1:61803 . . ., usually denoted by the Greek letter φ (PHI), has attracted broad attention for a long time, with suggestions that they are aesthetically significant and ubiquitous in nature. These claims were made prominent in Dan Brown’s book The Da Vinci Code: Despite PHI’s seemingly mystical mathematical origins, Langdon explained, the truly mind-boggling aspect of PHI was its role as a fundamental building block in nature
We introduce a scheme for population dynamics for analysable self-replicating chemical reaction systems, using system Eq (3) as an example
Summary
Introduction pffiffiThe golden ratio ð1 þ 5Þ=2 1⁄4 1:61803 . . ., usually denoted by the Greek letter φ (PHI), has attracted broad attention for a long time, with suggestions that they are aesthetically significant and ubiquitous in nature. . ., usually denoted by the Greek letter φ (PHI), has attracted broad attention for a long time, with suggestions that they are aesthetically significant and ubiquitous in nature. These claims were made prominent in Dan Brown’s book (and later film) The Da Vinci Code: (quote) Despite PHI’s seemingly mystical mathematical origins, Langdon explained, the truly mind-boggling aspect of PHI was its role as a fundamental building block in nature. Other examples include the design of the Parthenon, Leonardo da Vinci’s paintings, the ratios in a human body, lengths of passages in the Aeneid
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