Module 7

Systems Biology & Networks

Systems biology treats cells as networks of molecular interactions. Protein- protein interaction (PPI), gene-regulatory, and metabolic networks each have characteristic topologies and dynamics. This module covers scale-free networks, community detection, Boolean models, and flux-balance analysis.

1. Network Topology

Biological networks typically exhibit scale-free degree distributions P(k) ∝ k^−γ with 2 < γ < 3, following the Barabási- Albert preferential-attachment mechanism. Hubs (high-degree nodes) are often essential genes. Small-world structure — short average path length with high clustering — is another common property (Watts & Strogatz 1998).

2. Boolean & ODE Models

Boolean network models (Kauffman 1969) assign each node a binary state (on/off) updated synchronously by a logic function of neighbours. Attractors of the Boolean dynamics correspond to cell types or phenotypes. ODE models add continuous dynamics: Michaelis-Menten kinetics, Hill-function activators, and cooperative repression produce switches, oscillators, and feedback loops (Alon 2007).

3. Flux-Balance Analysis

For a metabolic network with stoichiometric matrix S (metabolites × reactions):

\[ \text{maximise}\ c^T v\quad \text{subject to}\quad S v = 0,\ v_{lb} \leq v \leq v_{ub} \]

The steady-state constraint S v = 0 and bound constraints define a feasible flux polytope; linear programming selects the flux distribution maximising a biological objective (biomass production, ATP yield). COBRA Toolbox (Python/R) is the standard. Genome-scale models (iJO1366 for E. coli, Recon3D for human) enable strain design and drug-target prediction.

Simulation: Scale-Free Networks & FBA

Python

script.py48 lines

import numpy as np, matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Scale-free network: Barabasi-Albert preferential attachment
rng = np.random.default_rng(0)
N = 200
m = 3
# Degree distribution k ~ k^-3 for BA network
# Simplified: sample from power law
k = np.round((1 - rng.random(N)) ** (-1/2.0) * 2).astype(int)
k = np.clip(k, 1, 40)
# Bin for histogram
bins = np.arange(1, k.max()+2)
hist, _ = np.histogram(k, bins=bins)

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5), facecolor='#0a0a1a')
for ax in (ax1, ax2):
    ax.set_facecolor('#111827'); ax.tick_params(colors='#cbd5e1')
    for s in ax.spines.values(): s.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.3)

ax1.loglog(bins[:-1], hist, 'o', color='#2dd4bf', markersize=8, markeredgecolor='#5eead4')
ax1.set_xlabel('Degree k', color='#cbd5e1')
ax1.set_ylabel('P(k)', color='#cbd5e1')
ax1.set_title('Scale-free degree distribution P(k) ~ k^-3',
              color='#5eead4', fontweight='bold')

# FBA stoichiometric matrix example
# Simple 3-reaction, 2-metabolite network
S = np.array([[ 1, -1,  0],     # metabolite A
              [ 0,  1, -1]])    # metabolite B
c = np.array([0, 0, 1])         # maximise flux of reaction 3
# LP feasible region visualisation (2 variables v1, v2)
v1 = np.linspace(0, 10, 200)
v2 = v1   # S v = 0 -> v2 = v1, v3 = v2 = v1
ax2.plot(v1, v2, color='#fbbf24', lw=2.5, label='Feasible flux ray (S v = 0)')
ax2.fill_between(v1, 0, v2, alpha=0.2, color='#fbbf24')
ax2.set_xlabel('v1 (uptake)', color='#cbd5e1')
ax2.set_ylabel('v2 (internal)', color='#cbd5e1')
ax2.set_title('Flux balance: S v = 0, max c^T v',
              color='#5eead4', fontweight='bold')
ax2.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
print('Barabasi-Albert: P(k) ~ k^-3 scale-free structure')
print('FBA: linear programming over metabolic stoichiometric matrix')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

4. Community Detection

Louvain and Leiden algorithms partition networks into communities by optimising modularity. Biological interpretations: PPI communities often correspond to protein complexes or functional modules; GRN communities correspond to co-regulated gene clusters. Cytoscape is the canonical visualisation tool; igraph (Python/R) and NetworkX are programmatic libraries.

Key References

• Barabási, A.-L. & Oltvai, Z. N. (2004). “Network biology: understanding the cell’s functional organization.” Nat. Rev. Genet., 5, 101–113.

• Alon, U. (2006). An Introduction to Systems Biology. CRC Press.

• Orth, J. D., Thiele, I. & Palsson, B. (2010). “What is flux balance analysis?” Nat. Biotechnol., 28, 245–248.

• Blondel, V. D. et al. (2008). “Fast unfolding of communities in large networks.” J. Stat. Mech., P10008.

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