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Wolfram Summer School

June 24–July 14, 2018
Bentley University, Waltham, MA

Alumni

Enrique Barrajon

Class of 2012

Bio

Enrique Barrajon is a Spanish and European certificate in medical oncology. He holds a PhD in medicine from the Universidad de Navarra and a master's degree in statistics and research design from the Universidad Autónoma de Barcelona. He is head of medical oncology at the Hospital Clinica Benidorm in Spain and a member of several national and international medical societies.

He is interested in mathematical modeling of cell growth, differentiation, mutagenesis, carcinogenesis, tumor development, immunology, clonal evolution, drug resistance, pharmacokinetics, and pharmakodynamics. Other interests are painting, mountain biking, and swimming.

Project: A Network Model of Cell Growth

Growth of biological systems has been modeled for nearly two hundred years, from the single exponential and the Gompertz model to generalized models as the ones proposed by Richards, Turner, or Tsoularis. Although continuous models are able to fit data and have advanced the scientific knowledge (the Gompertz model predicted, for example, the efficacy of adjuvant chemotherapy, proven years later and considered an standard of care in cancer patients), those models have the disadvantage of a poor interpretation of their parameters and the consequent failure to predict results.

Discrete systems may allow the modeling of discrete components with more resemblance to real elements. In the case of cell growth, a discrete modeling may differentiate between systems with unlimited growth, such as microorganisms, forests, or fisheries, only limited by the environment carrying capacity and the growth of organisms, reaching a limited size supposedly by feedback mechanisms.

In this summer course, I will use a minimal Boolean network to approach the growth of a protocell, taking into account some of the biological constraints already known. The basic idea is to model a biological cell using minimal gene networks.

The simplest network has just two nodes, where each node represents a theoretical gene (and its associated protein), cluster of genes, or pathway. Gene A (with protein a) controls DNA replication (e.g. gene POL), and gene B (protein b) controls cell division (e.g. gene MPF). The genes interact via their proteins. A protein can promote, inhibit, or do nothing to a gene (in reality, this occurs because the protein binds to the DNA near the gene, either up-regulating or down-regulating the rate of transcription of the gene).

Each time step of the model is one complete cell cycle. There are three phases in the cell cycle: (1) protein translation, (2) DNA replication, and (3) cell division.

In phase 1, the genes are translated into proteins. When gene A is translated, it creates another unit of protein a. However, protein b may inhibit this translation, causing no protein a to be produced (we say gene B inhibits gene A). Or, protein b may promote translation of protein a, causing an extra unit of protein a to be produced (gene B promotes gene A). In addition, protein a may promote or inhibit itself (a self-loop in the network).

We represent the gene network by an interaction matrix between gene A and gene B. The matrix has 4 entries: AA, AB, BA, and BB. Each entry has 3 possible values: promote (+1), inhibit (-1), or do nothing (0). There are 3^4=81 possible interaction matrices, or rules, in total.

During the translation phase, the number of proteins is represented by the vector {a,b}. The new population of proteins is given by {a,b} -> M.{a,b}, where M is the 2x2 interaction matrix, a and b are the protein populations (a more pragmatic approach than to build Boolean truth tables).

In phase 2, the genes are replicated, causing the number of gene A and gene B to double: A->2A and B->2B. However, recall that gene A controls DNA replication, so if A=0, replication will not proceed.

In phase 3, the cell divides, and half of gene A and gene are given to each daughter cell: A->A/2 and B->B/2 B (and the same for proteins a and b). But since gene B controls cell division, if B=0, the cell will not divide. If the cell divides successfully, the cell count N doubles: N -> 2N.

The goal of the project is to enumerate all possible interaction matrices (gene networks, rules in NKS) and exhaustively test their resulting dynamics, looking for cell growth N(t) that satisfies several constraints. First, the number of genes A and B must remain constant (constant ploidy). If cell division happens (B!=0) but DNA replication does not (A=0), there will be too few genes {A,B}, and the constant ploidy condition will be violated. Second, the number of cells N(t) should grow exponentially at first, but then it should level off (a sigmoid curve). If N(t) grows in an unbounded fashion, the organism will die (cancer).

The goal of the project is to enumerate all possible interaction matrices (gene networks, rules in NKS) and exhaustively test their resulting dynamics, looking for cell growth N(t) that satisfies several constraints. First, the number of genes A and B must remain constant (constant ploidy). If cell division happens (B!=0) but DNA replication does not (A=0), there will be too few genes {A,B}, and the constant ploidy condition will be violated. Second, the number of cells N(t) should grow exponentially at first, but then it should level off (a sigmoid curve). If N(t) grows in an unbounded fashion, the organism will die (cancer).

Favorite Four-Color, Four State Turing Machine

Rule 902500327781641807200256