Les cahiers de la Nuit ENS #6

Par Aleksandra Walczak et Thierry Mora,

Formally, uncertainty can be harnessed by reasoning in terms of probabilities. Unfortunately, as humans we do not intuitively grasp probabilities. We want certainty. Probabilities, tell us how likely each event in a list of outcomes is. They tell us there is a 60% chance of rain. How can this help us plan our hike or outdoor party? We painfully discovered with the COVID pandemic that the idea of exponential growth is often unintuitive:  how come everyone around me tests positive when last week no one I know did? Probabilities are even harder. Nevertheless, we need to grasp them to understand the future evolution of the virus, the weather, our long-term finances and everything that goes on in our body.

The secret to harnessing uncertainty is to develop baseline probabilistic expectations. This simply means figure out what to expect if nothing special happens. For example, if you live in Brittany, your baseline expectation for rain should be very different if you live in Marseille. It can rain in both, but if it rains every day in August in Marseille you would be surprised and start looking for reasons for this unusual event (which now you can call unusual). Whereas in Brest, well, you expect that. Having this baseline allows us to gauge the level of our surprise. Since stochasticity plays an important role at many scales in biology, from the molecular intrinsic noise in gene expression, through the rare events shaping evolution to the statistical description of cellular motion or animal groups, our intuition is often challenged. Here are two very different, yet very similar examples of how probability runs our lives. Birthdays and lymphocytes (white blood cells).

Birthdays

For each of us, our birthday is a special day. It is OUR birthday. At school teachers often spared us questioning on this special day. Although, as someone who took their last baccalaureate exam on their 18th birthday, unfortunately no one ever got out of any real test this way. Nevertheless, we feel special on our day (grumps who do not celebrate birthdays excluded). Have you ever wondered how many other people share the same birthday as you?

Perhaps the most elementary example of how our uneducated guesses fail when faced with probabilities is the so-called birthday problem. We are surprised that in a group of randomly chosen people, such as a classroom of children or a conference, two people have the same birthday. In fact the probability that at least 2 people among n people share the same birthday is given by 1-365!/(365-n)!/365n, which reaches 0.5 for only 23 people, and 99.9% for 70 people. So in a classroom it is likely that two people share their birthday, and it is nearly certain at a mid-size conference. Look around you? Are you in still at the Nuit de ENS? In the Ernests’ courtyard? In a classroom? Where is that person who shares your birthday?

It turns out that a lot of surprise can be easily eliminated, and with it the effort in searching for a deeper explanation (what do these two people have in common?), by developing such theoretical understanding. Yet in some cases, simple counting arguments are not enough and we have to actually work on a physical understanding of our expectations.

Lymphocytes

A generalization of the birthday problem presents itself when looking at the lymphocyte receptors shared by different individuals. The adaptive immune system uses a diverse set of specific receptors on the surface of B and T-cells to discriminate foreign pathogens from the organism's own proteins and trigger an immune response. The ones on B cells can also be secreted into the serum and are known under the more familiar name of antibodies. The correct functioning of the immune system relies on the diversity of these receptors. The composition of receptors, called the immune repertoire, evolves throughout our lifetime to reflect the pathogenic challenges we encounter. Thanks to high throughput immune sequencing of the proteins that make up human lymphocytes we can now compare the repertoires of different people and we find that they share a non-negligible numbers of receptors in functional repertoires, regardless of their lifestyles, environmental factors and family ties.

A calculation similar to the birthday problem, which corrects for the biased generation of immune receptors, shows that in such large ensembles of cells as the B or T cell repertoire, which contain upward of 108 unique receptors, we simply expect the most common receptors with a high probability of generation to be independently generated multiple times in different people. Knowing the bias of the receptor generation process (which requires to analyze sequence repertoires with advanced statistical techniques) shows that the number of shared unique receptors in healthy people can be almost entirely attributed to this convergent generation, and not to any functional response. Conversely, outlying receptors from this prediction are possible candidates for condition specific responding receptors, such as response to the SARS-CoV-2 virus.

Yet identifying common, even unusual receptors does not necessarily mean they are responding to the same pathogen. The immune response is a complex phenomenon that involves the binding of parts of the pathogen (called “epitopes”) to immune receptors, followed by signal propagation, cell commitment, and cell-cell communication through messenger molecules. By considering the phenomenological side of recognition, we can build a theoretical expectation of whether we expect two people in the same environment to have a similar distribution of receptors. Immune receptors are cross-reactive: one receptor can recognize many epitopes. Conversely, each one of these epitopes can be recognized by many receptors. This degeneracy is enough to generate many different, similarly well-protecting repertoires in different individuals, even if they sample essentially the same pathogenic environment. As a result, we should not be surprised by great differences at the phenotypic level, let alone the sequence level when looking at repertoires responding to specific threats.

We see that we need to educate our expectations. On the one hand, we now expect to see overlap in randomly chosen receptors by pure chance. On the other hand, we should not be surprised by a lack of overlap in responding receptors, because of the degeneracy of the recognition process. The examples we discussed show how a theoretical framework helps us to gauge our surprise, and to discriminate in which cases some behaviour is actually surprising (requiring us to explore some new phenomena) and in which cases the same behaviour is not. With a little work, we can build some probabilistic intuition and be better prepared for the unknown.