Can we say that the frequency distribution of each base in a DNA sequence is equiprobable?
After the negative answer; i rephrase: Is there a use case in which the frequency distribution of the bases is equiprobable?
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The frequency of the bases in the genome isn't equal to 0.25, the frequency depends on what kind of organism you mean. However let's take a look at some of them:
bacteria, most of the time we can see a bias towards some bases, this could be a GC bias, for example if the bacteria lives in extreme conditions, because GC can vorm three hydrogen bonds compared to AT which can only form two hydrogen bonds. So it's genome is more resistent to extreme temperature, less denaturation chance because the stronger bonds. In bacteria there are also some polymerases which have a bias towards bases, I won't discuss this in depth to keep it short, but from these two points we can already conclude the genome isn't random
Homo Sapien, for example we need CpG Island near our promotor region, to regulate transcription. Further we have higher GC% in gene coding regions to reduce mutation chance, because we don't want to mutate genes wich are needed for survival. Their are also some sequence which are very specific for example TATA box and Poly-A signal, if our genome would be random, so all base would have a frequency of 0.25 these specific sequence could occur at place where we don't want them or they wouldn't occur at places where we want them. We need a promotor (with TATA box) for RNA polymerase binding, if our genome was random we have a big chance we would have genes wihtout a TATA box so no transcription of these genes will occur
There are a lot more properties of the genome we could discuss but I think this will give you enough insight why the bases in the genome don't have a equally alike frequence of 0.25
in addition to your second question: I think the chance of finding a genome which has an equally alike distribution is very small. because almost all organisms have undergo evelution, and when we have evolution there is natural selection and natural selection relies on mutations. So (almost) all genomes have udergo mutation, however this sometimes can favor an equally distibution for example if our genome is AGAA and one A mutates to an G we will have an equally alike proportion. So to summarize: maybe you can take a look at NCBI and look for a gc 50% wich means 50% AT so 25% of each base. If you find a genome wich has exact 50% gc, you have found your answer
There are two main driving factors to consider here:
First, G/C and A/T represent complementary base pairs, as G is always paired with C (and vice versa), while A is always paired with T (and vice versa). Therefore the frequency of G is always equal to C, and the frequency of A is always equal to T.
Second, G is paired with C via 3 hydrogen bonds, while A is paired with T via 2 hydrogen bonds. Therefore, G/C bas pairs are more thermodynamically stable than A/T base pairs, and require higher temperatures (more energy) to be pulled apart during replication and also transcription. For that reason, organisms that have a higher body temperature tend to have a higher proportion of G/C base pairs than those which have a lower body temperature. For example, the G/C content is higher in thermophilic bacteria than in mammals, and higher in mammals than in plants. This fine-tuning of the G/C content enables the optimum balance between stability and the extra energy required to pull the strands apart when required for replication and transcription.