Biological Modeling: A Short Tour

Welcome!
Meet the Team
Acknowledgments
Random Walks and Turing Patterns
Alan Turing and the Zebra’s Stripes
An Overview of Random Walks
A Reaction-Diffusion Model Generating Turing Patterns
From random walks to a reaction-diffusion system
Parameters are omnipresent in biological modeling
Changing reaction-diffusion parameters yields different Turing patterns
Turing's patterns and Klüver's hallucinogenic form constants
A Coarse-Grained Model of Particle Diffusion
Diffusion of a single particle
Slowing down the diffusion rate
Adding a second particle to our diffusion simulation
Visualizing particle concentrations in an automaton
The Gray-Scott Model: A Turing Pattern Cellular Automaton
Adding reactions to our diffusion automaton
Reflecting on the Gray-Scott model
Conclusion: Turing Patterns are Fine-Tuned
Exercises
Solar photons and random walks
Practice with the cellular automaton model of diffusion
Changing the predator-prey reaction
Adjusting Gray-Scott parameters
Motifs in Transcription Factor Networks
Networks Rule (Biology)
Transcription and DNA-Protein Binding
The central dogma of molecular biology
Transcription factors control gene regulation
Determining if a transcription factor regulates a given gene
Transcription Factor Networks
The transcription factor network of E. coli
Loops in the transcription factor network
Gene Autoregulation is Surprisingly Frequent
Comparing a real transcription factor network against a random network
Negative autoregulation is very frequent
The Negative Autoregulation Motif
Simulating transcriptional regulation with a reaction-diffusion model
Ensuring a mathematically controlled comparison
The Feedforward Loop Motif
Feedforward loops
Modeling a type-1 incoherent feedforward loop
Biological Oscillators
Oscillators are everywhere in nature
The repressilator: a synthetic biological oscillator
Interpreting the repressilator's oscillations
Noise is a feature of biological systems, not a bug
Conclusion: The Robustness of Biological Oscillators
Biological oscillators are robust by design
A coarse-grained repressilator model
The repressilator is robust to disturbance
Exercises
A short introduction to statistical validation
Counting feedforward loops
Negative autoregulation
Positive autoregulation
Replicating the chapter's conclusions with well-mixed simulations
E. coli’s Genius Exploration Algorithm
The Lost Immortals
E. coli Explores Its World Via a Random Walk
Bacterial runs and tumbles
Tumbling frequency is constant across species
Signaling and Ligand-Receptor Dynamics
Cells detect and transduce signals via receptor proteins
Ligand-receptor dynamics can be modeled by a reversible reaction
Calculation of equilibrium in a reversible ligand-receptor reaction
Where are the units?
Example steady state ligand-receptor concentrations
Stochastic Simulation of Chemical Reactions
Verifying a steady state concentration via stochastic simulation
The Poisson and exponential distributions
The Gillespie algorithm for simulating well-mixed reactions
Confirming steady state calculations with the Gillespie algorithm
A Biochemically Accurate Model of Bacterial Chemotaxis
Transducing an extracellular signal to a cell's interior
Adding phosphorylation events to our model of chemotaxis
Changing ligand concentrations leads to an internal change
Methylation Helps a Bacterium Adapt to Differing Concentrations
Bacterial tumbling stays constant for different attractant concentrations
Bacteria remember past concentrations using methylation
Combinatorial explosion and the need for rule-based modeling
Bacterial tumbling is robust to large sudden changes in attractant
Traveling up an attractant gradient
From changing tumbling frequencies to an exploration algorithm
Conclusion: The Beauty of E. coli's Random Exploration Algorithm
Simulating a bacterium's motion
Simulated strategy 1: standard random walk
Simulated strategy 2: chemotactic random walk
Comparing the effectiveness of our two random walk strategies
Why is background tumbling frequency constant across species?
Bacteria are even smarter than we thought
Exercises
How does E. coli respond to repellents?
Simulating a bacterium traveling down an attractant gradient
What if E. coli has multiple attractant sources?
Changing the reorientation angle to E. coli
Can't get enough rule-based modeling?
Analyzing the Coronavirus Spike Protein
A Tale of Two Doctors
The world's fastest outbreak
Tracing the source of the outbreak
A new threat emerges
Protein Sequence and Structure
The sequence of the SARS-CoV-2 spike protein
Nature's magic protein folding algorithm
Protein Structure Prediction is Difficult
Experimental methods for determining protein structure
Protein sequence and structure do not correlate well
Flexible polypeptide chains can fold into many possible structures
Protein Biochemistry
The four levels of protein structure
Proteins seek the lowest energy conformation
Ab initio Protein Structure Prediction
Modeling ab initio structure prediction as an exploration problem
A local search algorithm for ab initio structure prediction
Applying an ab initio algorithm to a protein sequence
Homology Modeling
Using a known protein structure as a reference
Finding a similar structure reduces the size of the search space
Experiments determine the structure of the SARS-CoV-2 spike protein
Protein Structure Comparison
Comparing two shapes with the Kabsch algorithm
PDB format encodes a protein's structure
The Kabsch algorithm can be fooled
Applying the Kabsch algorithm to predicted structures
Distributing protein structure prediction around the world
Intermezzo: Did AlphaFold Solve the Protein Structure Prediction Problem?
Finding Local Differences in Protein Structures with Qres
Focusing on a variable region of interest in the spike protein
Contact maps visualize global structural differences
Qres measures local structural differences
Local comparison of spike proteins leads us to a region of interest
Analysis of Structural Protein Differences in the Spike Protein
Site 1: loop in the ACE2-binding ridge
Site 2: hotspot 31
Site 3: hotspot 353
Differences in interaction energy with ACE2
Gaussian Network Models (GNMs) and Molecular Dynamics
GNMs represent proteins using tiny springs
Representing random movements of alpha carbons
Inner products and cross-correlations
Mean-square fluctuations and B-factors
Normal mode analysis
Applying GNMs to compare spike proteins
ANMs account for the direction of protein fluctuations
Conclusion: Bamboo Shoots After the Rain
Exercises
Determining a shape's center of mass mathematically
Calculating RMSD by hand
Practicing ab initio and homology modeling
Trying out AlphaFold
Comparing protein structures with Qres
Calculating interaction energy
Visualizing glycans on the surface of SARS-CoV-2
Creating contact maps for the SARS-CoV-2 spike protein
Classifying White Blood Cells
How Are Blood Cells Counted?
Segmenting White Blood Cell Images
Image segmentation requires a tailored approach
The RGB color model
Segmenting an image based on a color threshold
An Overview of Classification and k-Nearest Neighbors
The classification problem and the iris flower dataset
From flowers to vectors
Classifying unknown data points with k-nearest neighbors
Shape Spaces
Stone tablets and lost cities
Vectorizing a segmented image
Inferring a shape space from pairwise distances
Aligning many images concurrently
Principal Components Analysis
The curse of dimensionality
How the curse of dimensionality affects classification
Dimension reduction with principal components analysis
Visualizing a white blood cell shape space after PCA
Classifying White Blood Cell Images
Cross validation
A first attempt at quantifying the success of a classifier
Recall, specificity, and precision
Extending classification metrics to multiple classes
Applying a classifier to a white blood cell shape space
Discussing limitations of our image classification pipeline
Conclusion: Toward Deep Learning
A brief introduction to artificial neurons
Framing a classification problem using neural networks
Defining the best choice of parameters for a neural network
Exploring a neural network's parameter space
Neural network pitfalls, Alphafold, and final reflections
Exercises
Neural networks and logical connectives
A little fun with lost cities
More on the curse of dimensionality
Irises, PCA, and feature selection
More classification of WBC images
Glossary
Appendix: Proof of the Random Walk Theorem
Image Courtesies