I don't think a priori you can make such a statement, since there's too many confounding factors. Most proteins in the cell have no idea whether they're in a liver cell, or a brain cell, or a pancreatic cell.
Well, all of them have "no idea", poor little things. But all that is necessary is that some mutation which is neutral with respect to one cellular environment should be harmful in another (in which it is also expressed, otherwise examples, while commonplace, would also be irrelevant).
For example,
No webpage found at provided URL: here's an account of the research of one Kevin Talbot:
The commonest and most debilitating of these diseases is amyotrophic lateral sclerosis, which is relentlessly progressive and fatal on average 3 years after onset. The key question for our research is why mutations in genes which are expressed in every cell in the body result in specific and selective degeneration of motor neurons. Using primary neuronal cell culture systems his group has explored the effect of mutant small heat shock protein on axonal function. Mutations in the gene for glycyl tRNA synthetase lead to another form of lower motor neuron degeneration. He is using a combination of structural, biochemical and cellular studies to understand why mutations in a ‘house-keeping’ gene expressed in all cells lead to specific neuronal degeneration.
It would seem, then, that the mutations he's studying would be neutral in an organism too primitive to have motor neurons.
Obviously cell environment has an effect on which mutations prove to be deleterious, because some of that effect is the result of changes in a protein's interactions with other components of the cell environment. But to predict the effect of diversity of cell environment on a mutated protein (and vice-versa) you need a finer-grained study than "liver cell; muscle cell."
Yes ... but we don't need to be able to predict the effects of a mutation to support the thesis that number of cell types correlates inversely with neutrality of mutations.
Well, it's a standard practice in constructing phylogenies to use proteins with identical function across the compared clades. That would seem to rule out any effect of diversity of cell environment.
The site quoted above specifically mentions glycyl-tRNA synthetase. This, as the name suggests, attaches glycine to transfer RNA with glycine antocodons. Hence glycyl-tRNA synthetase performs a very basic (and ancient) function --- which is the same function performed by glycyl-tRNA synthetase
in prokaryotes. And yet there are mutations to it which apparently in humans only produce problems with motor neuron cells.
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The questions remain: what is the magnitude of this proposed effect? And how on Earth are IDiots weaving this hypothesis into their tissue of nonsense? I don't see how this would help them any.