Series: Chasms of Evolutionary Impossibilities – Douglas Axe’s Work (2004) and the Evolutionary Impossibility of a Mere Protein.

doi:10.1016/j.jmb.2004.06.058

5. Probabilistic Analysis

5.1 Probability of Function

Douglas Axe's study (2004) did not just test proteins — it asked a profound question:

What are the chances that a functional protein arises by chance, without any guidance or design?

Axe tested billions of amino acid sequences and discovered that only approximately one in:

$$P(\text{function}) = (1.2 \pm 0.6) \times 10^{-150}$$

was capable of forming a functional fold — that is, a three-dimensional structure capable of performing a biological function. This number is incredibly small: a 1 followed by 150 zeros.

🪜 Analogy: Imagine trying to guess a 150-digit combination — and only one sequence works. The chance is smaller than finding a single specific grain of sand in all the deserts of Earth.

5.2 Resources Available in the Universe

To determine if this chance is feasible, we need to compare it with the physical resources of the universe. Let's consider:

Physical Resource	Estimate
Atoms in the universe	≈ \(3.4 \times 10^{80}\)
Time since the Big Bang	≈ \(10^{17}\) seconds
Operations per second (maximum)	≈ \(10^{15}\)

Multiplying all this, we get the total number of "possible attempts" the universe could make:

$$\text{Total resources} = 3.4 \times 10^{80} \times 10^{17} \times 10^{15} = 3.4 \times 10^{112}$$

🪜 Analogy: Even if every atom in the universe tried one combination per second since the Big Bang, it would still not be enough.

5.3 Comparing Chance with Capacity

Now we compare the chance of success with the total capacity for attempts:

$$P(\text{function}) = \frac{(1.2 \pm 0.6) \times 10^{-150}}{3.4 \times 10^{112}} \approx 3.5 \times 10^{-119}$$

That is, even if the entire universe were trying to generate a functional protein since the beginning of time, the chance of success would still be practically zero.

🪜 Analogy: It would require something like:

$$\approx 10^{119} \text{ universes like ours}$$

— each with all its matter, energy, and time — for there to be a single chance of generating one functional protein by chance.

5.4 Sequential Space and Number of Functional Sequences

For greater rigor, we can calculate the total size of the sequence space for a 150-amino acid protein. Since there are 20 standard amino acids, we have:

$$\text{Total sequence space} = 20^{150} \approx 10^{195}$$

Based on the functionality rate estimated by Axe:

$$\text{Functional sequences} = 10^{195} \times 10^{-150} = 10^{45}$$

🪜 Explanation for laypeople:
Even if there are \(10^{45}\) functional sequences — which seems like a lot — they are lost in an ocean of \(10^{195}\) possibilities. It's like having \(10^{45}\) needles scattered in \(10^{195}\) haystacks. The functional density is so low that finding one by chance remains practically impossible.

5.5 Calculation of Minimal Systems with Multiple Proteins

For minimal systems with multiple interdependent proteins — such as the blood coagulation cascade, which requires the precise interaction of 12 proteins — the improbability multiplies exponentially:

$$P(\text{system}) = (10^{-150})^{12} = 10^{-1800}$$

Considering the total resources of the universe:

$$\text{Resources required} = \frac{10^{-1800}}{10^{-112}} = 10^{1688} \text{ universes}$$

And if we include the evolutionary opportunity cost — that is, the time lost in non-functional mutations:

$$P(\text{system}) < 10^{-2000}$$

🪜 Visual analogy:
Imagine that each universe is a machine trying to assemble a functional watch with random parts. To assemble a single complete watch — with all gears working together — more than \(10^{1688}\) universes would be needed. This is more than the number of atoms in all the universes we could imagine.

5.6 Conclusion of Probabilistic Analysis

Even if this improbability were overcome by some extraordinary event, it would not solve the problem of evolution. Why?

• Because a single protein is not a functional system.
• It would need to be viable (non-toxic, energetically compatible), useful (integrated into an adaptive system), and preservable (capable of resisting genetic drift and deleterious mutations).
• Furthermore, it would have to be coordinated with other proteins, forming interdependent networks like those we see in real biological systems — for example, the blood coagulation cascade, which requires the precise interaction of 12 proteins simultaneously.

📍 Causal Dependency Map:

    graph LR
    A[Protein A] --> B[Functional system]
    C[Protein B] --> B
    D[Protein C] --> B
    E[Protein N] --> B
    B --> F[Irreducible complexity]
  

🪜 Explanation for laypeople:
Each protein is like a gear in a delicate mechanism. If one fails, the entire system stops. And no gear has an isolated function — they all depend on each other.

Doolittle (2013), an evolutionary source, admits:
“The evolutionary origin of the coagulation cascade remains unexplained by Darwinian mechanisms.”

5.7 Final Conclusion

The origin of functionality is not just a matter of molecular luck — it is a matter of intelligent causality.

The numbers do not just suggest — they shout — that life is a product of engineering, not chance.

With validated raw data, replicable calculations, documented self-refutations, complete causal mapping, and maximum falsifiability, this section reaches Level 0+ of the TDI hierarchy.

An insurmountable epistemic barrier for naturalistic explanations.