After trying to define the problem and find an answer mathematically, I remembered that I suck at statistical reasoning about as much as the average American. So I decided to model the problem with a short Python script and find the answer that way.

The problem: There are n people (say, at a party) drawn randomly from a population in which the chances of having a birthday on any day is equal to having a birthday on any other (which is not true of real populations (probably)). What is the probability of there being at least two people with the same birthday in the sample?

To put this thing together, I figure we need three things:

1. The ability to generate random numbers (provided by Python’s random module);
2. An object representing each person;
3. A party object full of those people.

Then we can add things like the ability to choose how many people we want at the party and how many parties to have, as well as some output for making plots!

First, the Person object. All each person needs is a birthday:

import random
random.seed()

class Person:
    def __init__( self ):
        self.birthday = random.randint( 1, 365 )

After that we need a Party object, which must be able to contain a bunch of people and check to see whether any of them have the same birthday:

class Party:
    def __init__( self, partiers ):
        self.members   = [ Person()        for p      in range(partiers) ]
        self.birthdays = [ member.birthday for member in self.members    ]
        self.check_matching_birthdays()

    def check_matching_birthdays( self ):
        birthday_frequencies = { b:0 for b in self.birthdays }
        for b in self.birthdays: birthday_frequencies[b] += 1
        for b in self.birthdays:
            if birthday_frequencies[b] < 2:
                del birthday_frequencies[b]
        self.matching_dates = len(birthday_frequencies)

Finally, we need the main() function allowing the user to choose the number of parties and attendees:

def main():
    while True:
        parties  = int(input("Number of parties:  "))
        partiers = int(input("Number of partiers: "))
        parties_with_matching_birthdays = 0

        for party in range( parties ):
            party = Party( partiers )
            if party.matching_dates:
                parties_with_matching_birthdays += 1

        print( 'Fraction of parties having at least one match:' )
        print( '   ', parties_with_matching_birthdays / parties )

main()

Putting it all together we get a simple program that allows you to answer the question given, and for any number of people. As you increase the number of parties, the variability gets smaller and smaller (and, being an inefficient program, the time taken gets longer and longer). The output I get from a few values is:

Figure 1: Frequency of finding individuals with the same birthday in groups of various sizes.

At 25 people there is already a better-than-even chance of finding two people with the same birthday, and by 50 it’s almost guaranteed! That does seem a little unbelievable, so I’ll refer you to the Wikipedia page on the topic for the math (yes, there really is a page for this).

Just to see if this little simulation would recreate what Wikipedia shows, we can modify the main() function and change it to the following:

def main():
    max_partiers      = 60
    number_of_parties = 10000
    iterations        = 3

    with open( 'birthdays.r', 'w' ) as rfile:
        # First add the variable containing each number of people.
        rfile.write( 'people = c( 1' )
        for partier in range( 2, max_partiers+1 ):
            rfile.write( ', ' + str(partier) )
        rfile.write( ')\n' )

        # Then the variables for the matches in each iteration.
        for iteration in range(iterations):
            rfile.write( 'matches' + str( iteration ) + ' = c( 0' )
            for partiers in range( 2, max_partiers+1 ):
                print( 'iteration =', iteration, 'partiers =', partiers )
                matches = 0
                for party in range( number_of_parties ):
                    party = Party( partiers )
                    if party.matching_dates:
                        matches += 1
                rfile.write( ', ' + str( matches / number_of_parties ) )
            rfile.write( ')\n' )

main()

Plotting the results in R gives:

Figure 2: Output of simulation. Plotted in R using the Cairo package.

And overlaying it onto the Wikipedia plot:

Figure 3: Overlay (using Inkscape) of simulated values (blue) on Wikipedia's calculated values (red).

]]> http://adamcoster.com/2011/07/13/python-shared-birthdays/feed/ 2 sacred atom party birthday simulation simulated_birthday overlay http://adamcoster.com/2011/07/11/cloning-trick-ligation-of-multiple-inserts/ http://adamcoster.com/2011/07/11/cloning-trick-ligation-of-multiple-inserts/#comments Tue, 12 Jul 2011 02:44:23 +0000 adamcoster http://adamcoster.com/?p=1184 Continue reading →]]> I’ve spent the last couple months building a plasmid library, and in the process I thought of a trick. Ligations, perhaps the worst part of cloning, are notoriously finicky reactions. The goal is to take several pieces of linear DNA, where the ends of the pieces can only connect in a certain way, and then use an enzyme (T4 Ligase) to sew them all together into one piece (in my case, a circular plasmid).

Figure 1. Ligase (2HVQ.pdb) rendered in PyMOL. Click to see a crappy animated GIF!

I needed to insert three fragments at once into a single backbone. In my ignorance (from my lack of experience) I thought ligating four fragments should work just as well as two, so I just threw them all together and ran the reaction. The result was a mess, and when I tested 40 different clones afterwards not a single one was correct. So I started adding them one piece at a time which, obviously, was going to take three times as long.

The next morning while in the shower (one of the best places for random ideas) I thought that, though most of what was in my test tube was not the desired product, there was most likely some tiny population that actually was the correct fragment. The problem was just that putting my products into E.coli and then trying to find the correct thing wouldn’t work if only 1/100 (or less!) of the ligation products were correct. But PCR is wonderful for making tiny amounts of DNA into large amounts, so I thought, why not just try to use PCR to get the correct thing out of the mess? Since each product is made out of some combination of the input fragments, the lengths would be discrete and therefore visible and, potentially, separable on a gel. I just needed to be able to copy up enough that I could see DNA run on a gel, and then I could just purify out the one that was the right length.

Figure 2. Each differently-colored fragment is being ligated into the gray backbone. The primers (black arrows) flank the entire insert.

I designed primers that would flank the combined insert and did a PCR (I can’t say “PCR Reaction” for the same reason I can’t say “ATM Machine”), and did indeed get a band of the desired length! After gel purification I was able to ligate the entire insert into the backbone without any trouble.

You might be asking, “Well isn’t this now a two-step process, just one less than the three required otherwise,  and so barely worth the effort?” Almost, except that there is no need to transform the multi-fragment ligation mixture. So it’s more like 1.5 steps. Plus, the ligation, PCR, gel purification, final ligation, and final transformation can all be done in a single day!

Though my labmates were surprised when this worked, it turns out I was a year too late to publish this idea . If you are fortunate enough to have access to Analytical Biochemistry (another journal that should be free to the public but isn’t), see the paper by An, Wu, and Lv called “A PCR-after-ligation method for cloning of multiple DNA inserts“. If you aren’t that lucky, I’ve already told you everything you need to know!

While I was working away at WashU, preparing for graduate school, I imagined myself as a grad student. In that imagination land I was working my ass off, learning all kinds of things, and sharing every bit of that with others via this platform. My work would be totally open. In reality, I am doing only those first two things.

Every couple of days I have an awesome research experience or an interesting new idea and come home planning to write about it. Then I start thinking about how I can present the experience while maintaining the proper level of censorship. Perhaps unsurprisingly, my motivation always quickly evaporates. You may be wondering: why is there any censorship required at all?

Well, it turns out that biological research must remain shrouded in secrecy for at least two major reasons: [1] Just like in any other profession, there are charlatans, thieves, and liars to avoid; and [2] biological research results are potentially worth huge mountains of cash. I’m still hoping that most biological scientists really do fit my ideal image of a Scientist, but the problem is that it only takes a few jerks to make life difficult for everyone else.

There is a lot at stake for researchers, as a lot of money and time is sunk into any project. So if some other group manages to do the same or similar work, and get published first, then much can be lost. Fear of outright idea-theft is one of the huge problems, but the more surprising one is that competing researches don’t want to give each other a leg up by sharing information because sharing leads to reduction or even loss of credit. If you share, your cool idea might end up becoming a novel tool used in someone else’s Nature paper instead of your own. And then you have more trouble getting funding, and students, and tenure…

So the funding system necessarily leads to closed science, which would be true even without the explicit jerks. And then money gets involved on the other side of discovery. Universities, companies, and researchers are patenting the crap out of new biological discoveries. Mutant proteins, dead viruses, synthetic compounds, techniques to synthesize or purify natural compounds, plasmids, and so on and so on. Most of these things aren’t worth much (except for knowledge-value) and so their defenses just end up making life a little more difficult. Others are hugely useful, but are better left to companies anyway (pharmaceuticals or complicated and expensive processes, for instance). Others still are hugely useful, but only for research purposes, and have no business being tied to business. For example: plasmids (circular DNA containing a handful of genes).

I have been building a plasmid for the past few weeks (for my secret research project) and ran into a totally unexpected problem. I wanted a certain fluorescent protein in my plasmid, and we have several in the lab on different plasmids that we have gotten from other labs or companies. For those unfamiliar with cloning (the molecular kind, not the sheep kind), it’s actually a rather routine process to stick a bunch of genes together into a plasmid. In order to get enough copies of a gene to efficiently construct one of these things, a researcher will use PCR. This awesome technology lets you go from a few copies of a gene to billions (literally) in a few hours. You can PCR any gene you want, so long as you know its sequence.

So my plan was to find the fluorescent protein gene from one of our plasmids, make a bunch of copies of it by PCR, then stick it into another one of our plasmids. Simple. But then it turns out that many of our plasmids  have Material Transfer Agreements (MTAs) attached to them, and that each MTA has different requirements for what I can and cannot do with the plasmid’s components. One plasmid has three genes, each “licensed” by a different university! One of the things the MTA prevents me from doing, for example, is to make any changes to the fluorescent protein gene from one of these plasmids. This actually sucks a lot, because that was something that I needed to do. And this requirement was by a university. I understand and expect that kind of behavior from a company, as companies exist to make money in whatever way possible. Companies aren’t supposed to have ethics, or have a primary goal of advancing human knowlege. But universities are supposed to have both of those things.

In the end, I just went with a probably-slightly-crappier fluorescent protein that, I hope, is not under an MTA. And it looks like it is doing what I designed it to do. So there’s that.

In conclusion, it looks like my relationship with biological research will become the same as my relationship with software: I’ll make my work as open as I can, hope that others do the same, and generally use free-and-open slightly-crappier versions of expensive-and-closed things (whether those things are plasmids or office software).

Hopefully the likely-more-complex reality turns around again so that I will find myself back at idealism. I expect that the process of publishing my first paper (whenever that happens) will reveal a lot…