## Statistics of the spontaneous generation of life

I am neither a scientist nor a mathematician, but I casually enjoy both fields. Please feel free to correct my math or science at any point, if you are an expert in the field. My goal is to understand the numbers behind the actual answer to this question: What are the odds of life spontaneously generating anywhere in the universe to date? I’m not expert, I assume I’ve made mistakes or left out variables. I hope I have enough knowledge to communicate my question to those who can reword it and answer it correctly, and that I have asked the right question.

*Weighing: the chance that life generates from a primordial soup of proteins.*

*Chance of life generating*

I assume the proteins are available. I assume that if the proteins line up in the right sequence, life immediately generates. How many combinations are allowed to create life (ie how many different arrangements are there in the DNA of single cell organisms, and how many correctly aligned pairs would it take for the simplest life?). **I’ll call the possible combinations to form basic life forms C and the minimum number of base pairs for life L.**

Assuming the average DNA strand is a volume of length*width*depth and in a specific space of S with the same volume, and proteins are moving through space S, each time a protein shifts into the space you would end up with a new “try.” (I’ll explain tries in the next section).

Each base pair sized location along S would correspond to a specific base pair needing to be in the right location and spatial orientation, there would only be 2 choices for each location as long as it was oriented properly. In a given space, each base pair would need to be oriented correctly within S or that try is null, so you’re comparing spatial orientation times the 50% chance of having the right base pair. For some conceptual simplicity I would propose that each combination of factors (X orientation, Y orientation, Z orientation, base pair type) is one “card” in a deck of cards. The number of cards in the deck would correspond to the possible combinations of X, Y and Z orientation and base pair type that could occur at any given base pair location.

*I relate this step to shuffling a deck of cards for each base pair location and pulling the correct card. If correct, continue on the next base pair location and repeat. If any pulls fail, the try is null, if all succeed, life is created.*

How far off in a 3d orientation can one base pair be before it is no longer recognized as a correct part of the system? If base pair 31 is in the position on the X Y plane, but tilted 15 degrees vertically, will it still operate in the strand? How about if it is tilted horizontally (still properly aligned along the DNA axis). The larger the margin of error is, the small the deck would be, because a possible DNA strand would be a compilation of all the DNA strands possible in the volume of error allowed. Also is there some chemical attraction or repulsion amongst base pairs that might cause them to align spatially more than random chance? If so, you’d have to have more qualifying cards or disqualifying cards based on the chance of proper alignment. **I’ll call the number of “cards” N.**

Also, can any be missing for the total strand to still be valid? If so, you’re allowed a certain number of failed pulls before you call the try a fail. **M will be missing base pairs/fails allowed+1.**

So, with life combinations C and base pairs for life required L, for this part of the equation you would take **M*C/N^L .That is the chance of occurrence on any given “try” (see next section).**

*Number of tries:*

**Space (starting big): **

A: How many stars are in the universe? (you can narrow your scope here depending on if you’re just calculating for life in the galaxy, our system, next system over, etc.)

B: How many earth like planets are there around the average per star?

(You could also just take the appropriate type of star(s) and then take their average earth-like count instead, might be more accurate)

C: What percentage of earth like planets would have the primordial soup present?

D: What is the average surface are of one of these plants?

E: What percentage of the surface would be covered in the soup?

F: What would the average depth of the soup be?

** ( A*B*C*D*E*F) is the total volume of primordial soup in the universe at one time, or P (for primordial).**

**Time:**

X : How many generations of stars have there been?

Y : How many generations of stars would it have taken before planets could have formed with the correct elements to create primordial soup?

Z : What is the average span of time that primordial soup stays on a planet (this might be a complicated relationship with E and F, if someone knows how to show soup on the planet as a smooth curve or time, we could use that number instead for more accuracy, for now I’m assuming the average for all 3 numbers)

**Z(X-Y) would be the amount of total time the soup would exist, or T**

Each primordial pool would have a number of similar and overlapping spaces to S, each oriented differently. If the pool was a perfect sphere with a diameter of the DNA strands length, there would be a number of spaces in the sphere of 4/3*pie*DNA length^3. (I assuming the number of these spaces would be divided by the width and depth of the strand in some way as well?).

**We’ll call this spherical shape with all possible DNA spaces of S included, Q.** We’d have to figure out how many Qs fit into a cubic meter to fit with the volume equation.

**So Q*P would be the number of spatial opportunities in the universe to create a working DNA strand in the entire universe. ** **Q*P*T would be the total potential tries in the universe to date in all of space time.**

**Velocity:**

Average velocity of the primordial soup would determine how many of the potential tries resulted in an actual try. If velocity were the ideal value, every potential try would result in a try (resulting in the full Q*P*T value). Since I’m guessing soup moved very slowly, only a percentage of the potential tries would be used. For example velocity might result in only 10% of the tries actually taken so the tries equation would be .1*Q*P*T.

Actually I don’t know how to begin on this one… it is the last frontier for me. How would the primordial soup move around? Wind, soup currents, earthquakes, heat vents? If anyone can come up with an average net velocity, and how that would translate into a percentage for the equation, or a way to eliminate this issue from the equation altogether, that would be excellent! Just picturing the scenarios, I imagine this number to be quite small, of how small. I’ll call this V.

**So as it stands V=actual tries taken/potential tries taken.**

**So the number of tries attempted in the universe to date would be: VQPT**

**The chance that life would have spontaneously generated anywhere in the universe to date would be:**

**MCVQPT/N ^{L}**

**Or: misses allowed in choosing a base pair*the number of different combinations allowed to create basic life*the percentage of actual ooze movement as compared to ideal movement*the total chances for a strand in a given volume of ooze*volume of ooze in the universe*Amount of time ooze has been around/(the number of “cards” to the power of how many base pairs are required to line up to form the simplest life forms). **

A few notes on this equation as I was writing it:

-Depending on what you’re looking for, this equation can be limited/expanded in a variety of ways:

You can use Earth’s value instead of trying to find average universal values for planet size, length of time with pools, etc. I will say this is likely to give a very generous number, as the fact that life exists here would suggest that our conditions for life are probably ideal (several standard deviations above the “average” planet that qualifies as earth like in the universe), so a balancing multiplier would also be prudent. We can also limit it to our galaxy, or to a particular nearby system. You can use just our star generation if you want to see what life might still be out there now (except for super intelligent beings from prior generations that escaped the death of their star).

-It doesn’t try to answer anything about what happens after life starts, and makes the rather silly (but very mathematically simplifying) assumption that once the DNA strand is correctly made, life exists. Feel free to make the argument for any modifiers you feel might be necessary for the formation of cell walls, viability of the life form long enough to begin replicating etc. I just thought this was already complicated enough.

-I would love feedback and constructive criticism. I’d also love it if anyone can provide their scientific or mathematical expertise to adjust the equation, or provide a specific number for any of the variables. If you have a number, please comment with the letter of the variable, the number you have, and your source or how you came to that number.

I was inspired to pose this question by my cousin’s retelling of a post he saw on Reddit (If you know of the post, please credit). Paraphrasing (in all cases) “What are you fairly confident has never happened on earth?” One answer: “Randomly shuffling a deck of cards back into its original order.” This apparently led to a discussion about combination math, compared to the vastness of space. It was said that “if the stars in the visible sky had multiple planets with the population of earth, with everyone was shuffling constantly since the beginning of time, you would just now be getting to the point where that might realistically happen once by now.” I didn’t believe it, I started doing the math, I was astonished to find the math worked out. That line of thinking started this puzzle brewing in my mind ever since.

-Evan