”This title is false”
Thus begins a reflection in the April 23 issue of Nature, by Mark Isalan and Matthew Morrison. They are alluding to Epimenides’ paradox, the famous philosophical concept where a statement, if true, must be false, and if false, must be true. (The paradox is usually related in the form: All Cretans are liars, as Epimenides himself was from Crete.) As Douglas Hofstadter so brilliantly exposed in his canonical book “Gödel, Escher, Bach”, this same phenomenon underlies Gödel’s incompleteness theorem. In effect, if a mathematical system is “complete” and able to incorporate such phrases as “this theorem is false”, it will therefore contain self-contradictions.
Statements referring back to themselves occur very frequently in biology, if you take a broad view of information transfer. For example, many cell signalling molecules induce an inhibitor of their own activity, in a negative feedback loop. What Isalan and Morrison highlight is the need to envision such systems not in the familiar two dimensions, but over space and time.
Consider the following ostensibly simple system:
It looks quite straightforward: p53 induces MDM2, which in turn inhibits p53. But can you predict how this system will behave?
A number of possibilities present themselves. Is there a steady state, where the two molecules are at equilibrium? Is there, perhaps, a constant slow decline of p53 until there is no activity left? It’s even imaginable that p53 could be completely uninducible in a system corresponding to the picture above.
In fact, p53 oscillates very dynamically when it is induced. It is thought that the number of peaks, rather than their amplitude, determine signalling intensity. This was only discovered fairly recently in Galit Lahav’s laboratory at Harvard.
The former president of the Karolinska Institutet, Hans Wigzell, often likens reaction pathways like the one above to “Donald Duck biology”. They present a tremendously oversimplified view of the system in question, by omitting time, space, and weighting of the processes involved.
With the rapidly increasing possibilities of resolving molecular interactions in time and space, we will have to get used to abandoning simplified models in favour of more complicated representations. I wonder how long it will take before differential equations are required knowledge for biology undergraduates?