Indeed it is! By combining data, we can predict how the antibodies in one dataset would inhibit the viruses from any other dataset. With this mindset, the different virus panels in each study are not a weakness, but rather an asset allowing us to better explore the space of possible viruses.

The implications are huge - each experiment takes significant time and resources, and by combining datasets we can massively expand the number of measurements, often by more than 10x, without any additional work!

Of course, things are not that easy. Biology is riddled with anecdotes highlighting how tiny differences in a protocol can lead to dramatically different results. Even without such errors, there is immense diversity among antibodies, so the patterns of virus inhibition seen in one dataset may not be reflected in another dataset.

To overcome this difficulty, we develop a machine learning algorithm to quantify the transferability of information between two datasets. In other words, how does the behavior of viruses in one study inform the behavior of viruses in another study? We display the transferability using a chord diagram, which is visually striking but a bit tricky to interpret the first time you see it. So let's look at one specific study...

Each band in the diagram represents a double-sided arrow, where a thicker band implies greater transferability. Study 1 is highly informative of Study 2 (and vice versa), whereas Study 1 and 3 are less informative of one another. Ferrets (shown in Study 7) turn out to be very different from humans, and we need to understand these relationships more deeply, since influenza surveillance is conducted using ferrets.

One surprising result was that human vaccination studies (shown in blue) and human infection studies (green) can have high transferability, even though the participants in the infection studies had never received an influenza vaccine in their life! By quantifying the transferability between datasets, we can determine which factors (such as age or infection history) give rise to distinct antibody responses, and which factors don't matter.

Of course, this is just the beginning. The power of this approach is that as we add more datasets, they are expanded by previous studies, but they also expand those studies in return. Thus, we grow a network that embodies the full extent of our knowledge on antibody-virus interactions.

This data-driven approach can be readily expanded to other viruses, or more generally to other low-dimensional datasets. I predict that the main limitation of these methods will be in how creatively we will apply them. If you have any ideas, or are simply excited by this approach, feel free to contact me and I would be happy to discuss this "mathematical superpower." You can also read our manuscript for more details.

To get myself oriented, I asked basic questions such as "How big is a flu virion?" and "How long does an infection typically last?" Regardless of the question, the most common response was, "It depends." I appreciated that answer - after all, humans are highly complex, and in my graduate studies I learned that even the behavior of individual bacteria can vary greatly, even though the population average was often tightly constrained. But when I pressed my colleagues for the typical range of values, no one seemed to know these numbers off the top of their head.

This response embodies a philosophical difference between biology and physics. Biology embraces the unique and breathtaking complexity of life, and the breadth of phenotypes is often of greatest interest. Physics promotes simple and unifying models, which can quantitatively describe key aspects of a system, but which might ignore some of its more nuanced behavior (think of the spherical cow analogy). Whether this tradeoff is worthwhile might be a matter of taste, but it explains why physicists yearn for key numbers in order to build quantitative models.

For example, notice that the influenza virion shown on the left is covered in teal and pink spikes. If you know the dimensions of the virus and the number of spikes, you could determine the nearest neighbor separation between adjacent spikes. Why is this an important number? It turns out that although antibodies often bind to the top of these spikes, the most sought-after antibodies bind to their stems (which are more conserved across flu strains). Given the average spacing between spikes, is there room for a stem-targeting antibody to bind to the stem of a spike?

Let's follow these questions through. The 2D pictures on the left show that viruses have a variety of shapes. Most of the viruses used in experiments are spherical, although the long and narrow morphology is believed to help a virus infect humans more efficiently. For spherical viruses, the mean diameter is typically 125 nm (including the lengths of the spikes on either end).

Given that there are 350 spikes spread uniformly on the surface of a virion, you can compute that the average separation will be 10 nm between nearest neighbor spikes. Since the width of an antibody is 15 nm, it will be a tight squeeze for it to bind to a spike's stem - it is possible, but the antibody will have to align itself just right. Indeed, squeezing into this tight slot reduces stem-antibody binding by about 10x.

This tight packing naturally leads to more questions. Is there enough room for every spike to be simultaneously bound by a stem-targeting antibody? If not, how are these antibodies able to stop a virus without binding to all spikes? Could it be that only a subset of spikes need to be bound in order to prevent the virus from infecting our cells? All of these questions are surprisingly understudied, and they are all motivated by some simple calculations that are only possible once we know some numbers.

To answer these types of questions, I read broadly on many aspects of flu biology, gathering numbers from all contexts. How many virions are produced during an infection and how many of our cells do they destroy? How many antibodies do we produce when we get infected by flu? How can we quantify the protection offered by our antibodies?

As I gathered these numbers, they naturally coalesced into different sectors of flu biology. As a fun mini-project, I put together a 1-page graphical illustration (published as a Cell SnapShot) that highlights the key numbers underlying an influenza infection and our antibody response against the virus. This SnapShot is targeted at influenza researchers who are specialists in one aspect of flu biology but would like a 30,000-foot view of some of its other characteristics.

One of the reasons I loved this project is that it melded my skill sets in science and illustration. Each of the figures was either created programmatically or was drawn from scratch in Adobe Illustrator. Feedback from my artistically-inclined colleagues led to me to think hard about the various aspects of graphical design from the iconography to the color palette.

As a fun example, the influenza virion in the center of the SnapShot was made in Mathematica, which meant that we could generate hundreds of variants and choose the one that looked the best. The pictures to the left display some of the most interesting virions that emerged!

As with any project, this one could not have been done without the help of many colleagues. I spent hours discussing the content and layout with Lauren Gentles, a terrific graduate student in our lab, and Jesse Bloom's feedback was instrumental in ensuring our scientific accuracy. You can find the final version of the SnapShot, representing all of our collective hard work, at this link.

We explored this question in the context of transcriptional regulation, when DNA is read in order to produce protein. Many aspects of this fundamental process have been extensively studied, and the main players are shown on the left.

Polymerase (light blue) binds to DNA and initiates transcription, the first step in transforming this segment of DNA into a protein. When more polymerase is present, transcription will happen more often, leading to increased protein production. To slow down expression of this particular gene, a cell can produce a repressor (red) which can only bind to this specific gene. The repressor competes with the polymerase for the same binding spots on the DNA, so that when the repressor is bound polymerase will not read this DNA.

But cells also respond to their environment, which introduces another level of regulation. A small molecule called an inducer (small blue circle) can bind to the repressor and inactive it, so that the repressor is no longer capable of binding to the DNA. The benefit of having these multiple layers of regulation is that it enables cells to respond on different time scales - cells can quickly change the local concentration of an inducer in order to inactive repressors in minutes, but reducing the actual number of repressors can take hours. One of the best studied examples of an inducible system (and the one that we ran our experiments on) is the Lac repressor in E. coli: the gene contains enzymes to digest the sugar lactose, but this gene is normally silenced by a repressor except when lactose (the inducer) is present.

While the interactions between these molecular players are qualitatively understood, there is no quantitative framework that can predict exactly how much gene expression will increase when the number of repressors are, say, doubled. Fueled by this challenge, we formulated a model for this system using a combination of statistical mechanics and thermodynamics. But we quickly ran into a problem - because this system was never quantitatively studied, many of the necessary parameters (such as how tightly the inducer binds to the repressor or the repressor binds to DNA) were unknown.

To that end, we took data on a single strain of bacteria in order to characterize all of the parameters of the system. After that, we had the keys to the kingdom. We could predict exactly how the system should behave in any other setting. For example, it is experimentally straightforward to tune the number of repressors in a cell. So using our single data set (orange circles), we predicted how other strains should behave. And then came the real question: will our predictions accurately portray the behavior of our system?

Before showing you the result, it is worth stressing once again that these predictions were made based on data from a single strain, before we constructed and measured any other strains. This is a prime example of theory leading the way to make strict predictions that are either carried out or disproved by experiment. This is quantitative biology at its best.

And for readers with a strong statistics background, the shaded areas represent the 95% credible region using Bayesian model fitting.

In total, we created 18 different strains of bacteria (6 of them are shown on the left) to test our framework under many different experimental conditions. In each case, the data matched remarkably well with the predictions!

This benign-looking data set represents a huge experimental effort. Each of the 72 data points is the average of eight different experimental runs, with each run analyzing over 40,000 cells. The whole procedure took several months to carry out experimentally - in contrast we fleshed out the theoretical framework over the course of a few weeks. That is not a dig on experiments. Rather, this difference in time scale demonstrates how important it is to be able to explore a system theoretically.

Amusingly, once we started to present these results to other professors, we began to get feedback such as, "Why don't the dark blue data points lie exactly on the line? They seem to be off by about 10%." And this led to an impromptu conversation among the professors in the audience, many of whom had the opinion that biology, being a nascent field, is a "factor of 2" science where you are lucky to have a theory that matches your experiment to within a factor of 2. As the field develops, more exact theories will surely arise that minimize the small discrepancies between our theory and the data, yet it should be appreciated that before our paper there was only a qualitative understanding of inducible gene regulation, and the plot on the left represents a significant boost in our ability to model such systems.

The fantastic consistency between our theoretical predictions and experimental measurement does not spell out the end of the story. Rather, it provides justification that our theory is correct, so that we can now begin to explore transcriptional regulation theoretically (which is orders of magnitude faster than exploring it experimentally!). We can now make quantitative statements about the effects of the physical parameters governing the system (such as the binding strength between the repressor and inducer) and explore all of the possible phenotypes that a cell can exhibit. The data shown on the left is just the tip of the iceberg. This experiment has paved the way for a truly exciting possibility of quantitatively exploring the effects of mutations in the context of transcriptional regulation (see below).

More details about this work, as well as the full data set and analysis notebooks (in both Python and Mathematica) may be found at this dedicated website.

There are many other molecular players involved in enzyme kinetics. One such player is a competitive inhibitor which fits into the same site as the substrate, but which cannot be cleaved by the enzyme. Hence, this molecule competes for the binding site and inhibits the reaction.

How will the speed of an enzyme depend on the amount of competitive inhibitor? This one is easy. The inhibitor blocks the reaction, so naturally the enzyme's speed will decrease as you add more competitive inhibitor.

And yet, the data disagrees! For some enzymes, adding a small amount of competitive inhibitor can increase the enzyme's speed. Ridiculous! By now your heart is really thumping, and you want some answers.

Before getting some answers, let's appreciate the context of these problems. The first phenomenon (dubbed "substrate inhibition") has been documented hundreds of times and is believed to occur in approximately 20% of all enzymes. This prevalence suggests that substrate inhibition may serve some critical function in enzyme behavior, and we also acknowledge that the known mechanisms and advantages of for substrate inhibition will strongly depend on context of a particular enzyme.

Let's focus on the second effect (called "inhibitor acceleration") which is comparatively rare in enzymes. Our model will be extremely simple, requiring only the three relevant molecular players: the enzyme, substrate, and competitive inhibitor...along with one other key ingredient.

Allostery is the concept that an enzyme (or any other protein) can exist in two different conformational states. For our model, we will assume that an enzyme can exist in either an active state (where it is able to cleave substrate extremely quickly) or an inactive state (where it speed is greatly diminished). The active state can often be hundreds of times faster than the inactive state.

Why these two allosteric states exist is an excellent biochemistry question. As a very rough explanation, proteins are very wobbly structures that often have access to numerous different configurations. Indeed, allostery is found in practically every biological context from ion channels to molecular motors to regulator proteins. In our simple model, we assume that an enzyme spends the vast majority of its time in only two of these states, so that we can neglect the other states.

Now for the solution! Consider an allosteric enzyme which flip flops between an active (fast) state and an inactive (slow) state. Suppose this enzyme has two identical and independent binding sites for substrate. (Enzymes often have multiple binding sites, as it permits them to further increase their speed.)

Assume that, in the absence of substrate and competitive inhibitor, the enzyme prefers to be in the slower inactive state. If we now add substrate molecules, the enzyme will lethargically convert them into product.

Now let's add competitive inhibitor into the mix. Assume that when the inhibitor binds to the enzyme, it forces the enzyme to assume the much faster active state. So what happens when we vary the amount of competitive inhibitor?

The two extremes are the easiest to determine. In the absence of any competitive inhibitor, the enzyme will be chugging along at the laid-back speed of its inactive state. In the other limit, if we completely saturate the system with competitive inhibitor, the substrate will never bind to the enzyme and hence the enzyme's speed will go to zero, as no product is being made. But the middle regime is the interesting bit. If there is just enough competitive inhibitor that only one of the enzyme's sites is bound to competitive inhibitor, then the other site is free to bind substrate at the fast rate of the active state. If the speed of the enzyme in the active state is more than double its speed in the inactive state, then the fact that the inhibitor clogs up one of the two binding sites is made up for by the change in allosteric states. Mystery solved!

In working out the details of such system, the math not only provides you with precise conditions for when such peaks happen, but often times yield unexpected results. For example, the picture we painted above indicates that inhibitor acceleration only requires three assumptions: (1) an allosteric enzyme that (2) naturally favors the inactive state interacts with (3) a competitive inhibitor which favors the active state, then a peak should form. Yet when you carefully work through the theory, crossing the t's and dotting the i's, you find that one of these assumptions is actually not needed. You can find the juicy details in this paper, which I wrote with Linas Mazutis and Rob Phillips. A poster summarizing this work was presented at ASCB 2016.