#15 - useful models of relationships
useful models. a story of paxlovid rebound. linear vs exponential. lennard jones potential as a model of relationships. critical behavior. love.
I have wanted to start a podcast called useful models with Chris, where we would discuss random topics and try to apply mathematics or physics to explain them. The name comes from Chris’ frequent evocation of Einstein, possibly apocryphally:
“All models are wrong, but some are useful.”
It’s a reminder to not over bias on rationality while not desperately giving up trying to understand in the face of complexity and uncertainty either.
Models are, mathematically, compression algorithms; they store information efficiently, but details are lost in that density. For example:
The blue dots represent data points with each data point requiring two values to store, an x-axis and y-axis value. There are 18 total data points, each with two values and together require storing 36 independent values. The line of regression, a line that is used to model or “best fit” the 18 data points, requires just two values: the value of the y-intercept and the slope of the line. The model is easier to store than all the individual data points. But if you were to scrap the data and just keep the model, it is impossible to completely recreate all the individual data points from the model, and the model is thus lossy. With better models, more information is efficiently retained.
A Useful Model: Exponential vs Linear Effects in Paxlovid Rebounds
We first thought about this when I was considering a question posed by my mom:
“Why does paxlovid rebound occur, when after a few days of COVID negativity, you test positive again?”
No one knows for certain, and neither do I. Some people hypothesize that there are tissue reservoirs of virus that drugs are unable to access or mutations in the virus. It sounds plausible but there is simpler intuition regarding this and it involves modeling the differences between linear and exponential effects.
Viral infection is exponential: one virus infects a cell, uses the cell’s machinery to replicate and when the cell dies, it releases, let’s say, ten viruses. These ten then infect 10 other cells, and that becomes 100 viruses, and so forth until the virus becomes a full blown infection. It’s like the story of the single grain of rice that doubles each day, which leaves someone with over a billion grains of rice after just one month, illustrating the power of exponential effects.
Paxlovid works by inhibiting two different enzymes that the virus requires for its activity. Drugs, though, work more or less on a linear basis. You take a finite number of molecules (300 mg nirmatrelvir with 100 mg ritonavir per day) and they can do their thing as long as they’re in the body. But eventually those molecules get metabolized and excreted by the body. Taking multiple doses of the drug is additively linear. One molecule of drug can only work on one enzyme at a time; they don’t replicate like the grain of rice or viruses.
Unless paxlovid rids every last infective virus in the body, eventually exponential viral replication will outrun the linear inhibitory effects of the drug. But why doesn’t the virus eventually over take us?
The answer is the immune system. The immune system responds to an infection exponentially, as well. Immune cells, once activated replicate to quickly and potently match and eventually outrun an infection. In fact this exponential effect of immunity is so powerful that it has been documented that a single T-cell (a CAR-T, I might add), can and has eliminated almost seven pounds of leukemia (~ 3 billion tumor cells) in a patient in less than two months. But the immune system lags the infection, typically by a few days, needing time to observe, assess, activate, proliferate and then resolve.
Without paxlovid, testing for COVID would yield the well known period of positivity, days after the initial exposure, followed by resolution, as shown in the graph below.
But with Paxlovid, it would look like this (not drawn to scale or with any scientific rigor, to be fair), with a rebound following a negative test:
Thus a simple linear/exponential model can describe the Paxlovid rebound. QED. Which by the way, this might not explain it at all! All models are wrong, right? But it’s a possibility and perhaps a useful one at that.
A Useful Model: The Lennard Jones Potential to describe human relationships
So besides being a potential band name for a 1970’s era cover band (thank you Tim Colbourne!), the Lennard Jones Potential (LJP) is a physics/math model for the interaction between atoms, molecules or colloids. For people work in such areas as statistical mechanics or molecular dynamics simulations, they use LJP to understand the condensed matter physics of micelle formation, liquid crystal self-assembly and viscoelastic fluid behavior. But at its heart, LJP is quite simple: it is a balance of attractive forces and repulsive forces.
An example of LJP: say you have a couple atoms:
Without getting into the underlying physics of London dispersion forces (dipole-dipole interactions) or Van Der Waals effects, the LJP basically asserts that in the balance of attraction and repulsion there is a goldilocks spacing between atoms. The trough, in the graph below, is the lowest potential energy state, the equilibrium, or most likely distance between atoms, which can be closer or farther apart, or deeper in intensity, depending on the relative amounts of attraction and repulsion.
But beyond some nerdy description of intermolecular interactions, it is actually quite a beautiful metaphor. That in the inherent fundamental behavior and physics of all substance and matter, there are just two competing forces acting on it all:
attraction - our desire to be with one another and to coalesce and bond
repulsion - our desires to be our own individual selves
We can apply LJP as a model for human relationships. Attraction is what we feel for each other, that impulse to be close with one another. Repulsion is the feeling of, not anti-attraction, but rather the need to value the self, this ineffable notion that we are discrete and disparate from one another. Too far or lonely, we desire connection (attraction), but too close it feels like annoying trespass (repulsion). At optimum, depending on the strength of the attraction and repulsion, we keep certain people close to us in relation and others at a farther distance This LJP model seems to ring true of any relationship - romantic, friendship, collegial, neighbor or other.
Exponential impact of LJP on relationships
The way I’ve drawn the cartoon above, is that of two atoms forming an LJP equilibrium between them. But what of us existing in groups larger than just two? Humans exist in groups, tribes, clans, societies and nation states wherein we are intimately connected to one another. As people in a group increase, we realize that it’s not just one relationship between two members. Rather each individual harbors multiple relationships binding them all together. Quickly the number of relationships far outpaces the linear increase in the number of participants in the group and a superstructure rises, a network of relationships. Physicists explain this as a critical phenomenon1 which relies on interactions between individual parts. It is through these concerted interactions that we effect one to two and two, four, and so forth.2
This framework is quite interesting and hopeful. It shows that our relationships are what give humans exponential possibilities. If we were all repulsive to one another, i.e. individualistic, and thus far apart and isolated, we would never exceed the linear scale of the singular. Humans would merely be the sum of their parts. But because of our relationships, the ties that bind us, our interactions, love, we have the ability to be limitless! Our relationships serve as the fulcrum of exponential human potential. What’s even more mind-blowing, is that even if you remove one of the “atoms,” to death, or layoff, or to relocation, the residue of the relationship continues to exist, persisting in our memory and in the intangible substance of our collective being.
What could be more scalable than what we would otherwise called love? Isn’t love, our desire to be close to one another, limitless?
LJP tells us that the opposite of love is not hate, it’s indifference. The attractive force, or lack thereof, is the driver of connection. More importantly, the LJP model of human relationships explains why it is critical to invest in connection among our groups. Together, we can accomplish more. As we discuss company culture here, at Lessons in Chimeristry, this is why we believe that building connective, percolated communities within our companies are critical for getting outsized performance. This is only step one though. We still must get that entire community moving in the same direction. We’ll look further at LJP, possibly mathematically, in the future.
Metaphors as useful models
We will continue to touch on various useful models. I mean foursquare is a useful model too. But I wanted to end on one note about metaphors as models for our experiences and perceptions. Above I stated that models are compression algorithms that are information efficient, but lossy. But metaphors are different and special in that they are almost entirely wrong. For example, Shakespeare writes:
All the world’s a stage, and all the men and women merely players.
The world is most definitively not a literal stage and most are not players. But this line allows Whitman to then further build:
That the powerful play goes on, and you may contribute a verse
Metaphors are wrong because they transmute. They compress language. They are pregnant with uncertainty and polysemy. But they open up a vast, cavernous diastemic frontier where ideas, inspirations, insights and imaginations collide together, interconnect and blossom. Complexity flourishes in metaphor.
As do our relationships. After all it is our closest relationships that alchemize us into the subtle infiniteness of our being, whom we were always meant to become.
critical phenomenon are sometimes known as emergent or self-similar. Some examples of this are crack propagation, snowflake structures (fractals), percolation theory in polymer solutions, idea generation, phase changes like crystallization.
A good example of a 1 —> 2 —> 4, etc. phase change is the solid to liquid phase transition of an avalanche. One pebble loosens two, two loosen four and so forth until an entire side of a mountain side slides.