Noncommutative Biology: Sequential Regulation of Complex Networks and Connected Matter

Citation

Letsou, William Peter (2018) Noncommutative Biology: Sequential Regulation of Complex Networks and Connected Matter. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/9B5E-F105. https://resolver.caltech.edu/CaltechTHESIS:05292018-192944407

Abstract

During animal development from zygote to adult, a limited set of regulatory molecules are autonomously deployed in the service of tissue-specific gene expression (reviewed in chapter 1). Inherent in the process is the tension that single cells sample heterogeneous expression states while robustly maintaining a collective final outcome. This thesis addresses theoretical issues that help resolve the paradox that one cell simultaneously contains the fate information of many.

Previous models of development have likened cell fate to minima on a smooth potential energy surface. Such static pictures can be misleading because they suggest the egg knows the path it will take to the adult before it divides even once. Recognition that the potential analogy is an oversimplification has led others to propose that the surface is actually nonsmooth. Chapter 2 reviews the theoretical basis for smooth potentials and resolves these problems by appealing to the tangent space of gene expression. It is then shown that if the potential difference is sufficient to characterize the difference between egg and adult, then the tangent space controls on gene expression are one-dimensional. Furthermore, a shortcoming of models ignoring the connectivity and common origin of dividing cells is that they erect artificial barriers between alternative fates. A fundamentally different picture is sketched wherein the difference between egg and adult is schematized as the shape of the locus of equipotential fates accessible at the same point in time. The conjugacy of space and time is invoked to explain how the requirement that each fate be on a line of equipotential is the same as requiring that each alternative fate move the same distance down the surface at each step. The developmental trajectory is deterministic but not known in advance because it needs to be ascertained at each step which way cells "turn" in order to maintain their equipotential relationship. Chapters 3 and 4 refine this sequential model of collective development with specific examples.

A simple solution to the problem of cell-type specific gene expression is combinatorial binding of transcription factors at promoters. It is shown in chapter 3 that such models result in substantial information bottlenecks, because all cell fate information is concentrated at the start. We explore a novel, noncommutative model of gene regulation—known as sequential logic—that spreads the information out over time. It is shown using time sequences of noncommutative controllers that targets which otherwise would have been activated together can be regulated independently. We derive scaling laws for two noncommutative models of regulation, motivated by phosphorylation/neural networks and chromosome folding, respectively, and show that they scale super-exponentially in the number of regulators. It is also shown that specificity in control is robust to loss of a regulator. Consequently, sequential logic overcomes the information bottleneck in complex problems and enables novel solutions through roundabout strategies. The theoretical results are connected to real biological networks demonstrating specificity in the context of promiscuity.

Noncommutative sequential logic has improved storage capacity, but it does not specify who or what supplies the sequences of input that determine cell fate. Chapter 4 offers a solution by way of the seemingly unrelated problem of looping in twisted strings. Cells and strings obey a set of common space-time constraints, ultimately due to the conservation of energy. It is argued that the most parsimonious allocation of energy from the straight to strained string is the one in which each segment sees the same share of the total. Planar looping is shown to be a consequence of the parsimony principle and the Euler-Poincaré equations for rotational motion in the presence an applied torque. We then solve the problem for the looping of a twisted string; with two strains, the Euler-Poincaré predict a different answer than the classical Frenet-Serret equations. Using the results of chapter 2, it is concluded that the Frenet-Serret curvatures assigned ahead of time are not guaranteed to generate space curves that conserve energy: the predicted string has localized strains the Euler-Poincaré solution lacks. Rotational dynamics of strings are connected to developing organisms by postulating conserved RNA polymerase as an analog of angular momentum, and transcriptional activity as energy. Alternative fates along a one-dimensional "string" of dividing cells are possible by finding the RNAP distribution that conserves transcriptional activity along a curve of constant developmental potential. Consequently, each alternative fate samples a different sequence of changes to the distribution as it follows a local gradient downhill from high to low developmental potential over time.

In conclusion, regulation in the tangent space of gene expression resolves the paradox that development has a unique solution specified in the DNA of the egg which cannot be determined with certainty until completion of the adult. Noncommutative sequential logic generates complexity that cannot be realized at the start, while interdependent cells (and strings) require time to ensure that each fate is at the same potential difference from a common ancestor. This fundamental reimagining of the Waddington framework can be tested using new multiplexed mRNA imaging technologies that preserve the spatial context of cells in developing tissue.

Thesis Files