consider more than one type of particle are randomly walking

\([A]_{\text{new}} = [A] + \triangle A + f.(1 - [A]) - r.[A][B]^2\) \([B]_{\text{new}} = [B] + \triangle B - k.[B] + r.[A][B]^2\)

Applying these reaction-diffusion computations

a cellular automaton called the Gray-Scott model

very slight changes in parameter values can lead to significant changes in the system.

frequently recurring structures hidden within biological networks

find the set of genes to which a transcription factor binds

transcription factors and genes make a network

simulation

three components are transcription factors, and they repress themself in a loop

all three particles are produced as the result of an activation process by some other transcription factor

image activation of protein Z, X and Y should be transcription factors

we ignore X kill rate because the system produce it in a way that its concentration become steady

How does a simple organism like E. coli sense an attractant or repellent in its environment?

The movement of organisms like E. coli in response to a chemical stimulus is called chemotaxis.

attractants like glucose and electron acceptors

repellents like \(\text{Ni}^{2+}\) and \(\text{Co}^{2+}\)

Every E. coli cell has between five and twelve flagella distributed on its surface that can rotate both clockwise and counter-clockwise.

propel the cell forward at about 20 μm per second.

it is about ten times the length of the cell per second

alternate between periods of “running” in a straight line and then “tumbling”

variations in bacterial tumbling frequencies are relatively small.

In the absence of an attractant or repellent, E. coli stops to tumble once every 1 to 1.5 seconds, which is similar to most other bacteria.

we wonder why evolution might hold tumbling frequency constant across species.

perceive a change in its environment and react accordingly

cell identifies a stimulus outside the cell and then transmits this stimulus into the cell.

certain molecule’s extracellular concentration increases

have more frequent binding with these molecules

This signal is then “transduced” via a series of internal chemical processes.

cascade begins that eventually changes a transcription factor into an active state, so that it is ready to activate or repress the genes that it regulates.

receptor proteins that detect attractants such as glucose by binding to and forming a complex with these attractant ligands.

To model ligand-receptor dynamics, we will use a reversible reaction that proceeds continuously in both directions at possibly different rates.

if we have \(l_o\) from L and \(t_0\) from T the system should become equilibrium in some point

the rates of the forward
and reverse reactions are equal.

This mysterious new disease had crossed the Pacific Ocean within a week of entering Hong Kong.

SARS was deadly, killing close to 10% of those who became sick.

researchers published the results of sampling horseshoe bats for five years in a cave in Yunnan province. They found that these bats harbored coronaviruses with remarkable genetic similarity to the virus causing SARS.

their outer membranes are covered in a layer of spike proteins

both SARS-CoV and SARS-CoV2 has spike protein

related to several coronaviruses isolated from bats and distantly related to SARS-CoV

This folding process occurs spontaneously for all proteins and without any outside influence

the same polypeptide chain will almost always fold into the same 3-D structure in a manner of microseconds.

Determine the spike protein’s shape from its sequence of amino acids?

preserve thousands of copies of a protein in non-crystalline ice

Protein structures that have been determined experimentally are typically stored in the PDB, which contains over 160,000 protein structures.

fine-tuned with respect to some mutations but robust with respect to others

As a result, two very different amino acid sequences can fold into proteins with similar structure and comparable function.

hemoglobin subunit alpha taken from three species: humans, short-fin mako sharks, and emus.

the angle of this bond is fix but it's more flexible for C_alpha

So we can have different shape for a chain of amino acids

three chain -> three identical protein structures

the SARS-CoV-2 spike protein has a receptor binding domain (RBD) located on the S1 subunit of the spike protein that is responsible for interacting with the human ACE2 enzyme

tend to find these amino acids tucked away on the interior of the protein.

A central theme of Chapter 3 was that a system of chemical reactions moves toward equilibrium. The same principle is true of protein folding; when a protein folds into its final structure, it reaches a conformation that is as chemically stable as possible.

potential energy = free energy = bonded energy + non-bonded energy

attraction and repulsion forces from the electric charge between pairs of charged amino acids.

Two nearby amino acids of opposite charge may interact to form a salt bridge.

due to random chance, the negatively charged electrons in an atom could momentarily be unevenly distributed on one side of the nucleus.

one side of the atom may attract only the oppositely charged components of another atom

As the protein folds, it seeks a conformation of lowest total potential energy

\( \min_{structures~for~P} potential~energy~(strucrure)\)

Predicting a protein’s structure using only its amino acid sequence

sample n, \(s_i\) and \(t_i\) points from two shapes

the information contained in known structures to help us predict the structure of a protein with unknown structure.

Critical Assessment of protein Structure Prediction (CASP)

In 2020, the 14th iteration of this contest (CASP14) was won in a landslide.

The second version of AlphaFold, a DeepMind project, vastly outperformed the world’s foremost structure prediction approaches

Do the mutations that SARS-CoV-2 has accumulated somehow make it easier for the virus to infect human cells?

In addition to verifying the structure of the spike protein in both SARS-CoV and SARS-CoV-2, researchers also determined the structure of the RBD complexed with ACE2 in both SARS-CoV (PDB entry: 2ajf) and SARS-CoV-2 (PDB entry: 6vw1).

M(i, j) = 1 if amino acid i is more near to j than a threshold

hidden away from the outside of the ACE2 enzyme

we can compute potential energy for ACE2 and Spike protein Complex

this complex has fewer energy in ACE2 - SARS-CoV2 Spike binding domain

What about dynamic movements of protein structure?

Gaussian network model (GNM) is a kind of ENM

The fluctuation of carbons are Gaussian, meaning that the alpha carbon deviates randomly from its equilibrium position according to a normal (bell-shaped) distribution.

Although atomic fluctuations are powered by randomness, the movements of protein atoms are heavily correlated.

what is relation between \(\Delta R_i\) and \(\Delta R_j\) ?

\(C_{ij} = \frac{<\Delta R_i, \Delta R_j>}{\sqrt{<\Delta R_i, \Delta R_i> . <\Delta R_j, \Delta R_j>}}\)

B-factor = \(\frac{8 \pi^2}{3} <\Delta R_i, \Delta R_i>\)

deconvolving a protein’s movement into individual normal modes

a technician filters a small amount of blood onto a gridded slide

counts the number of cells of each type in the squares on the grid

identify and attack foreign cells as part of your immune system

divide into families based on their structure and function

\(RMSD(s, t) = \sqrt{\frac{1}{n} [d(s_1, t_1)^2 + \dots + d(s_n, t_n)^2]}\)

the longest line segment that connects two points on the outside of the image and crosses through the image’s center of mass.

The best known classification algorithms for WBC image

if we add more particles, solving equations become hard

how we find \(k_{bind}\) and \(k_{dissociate}\)?

on average, \(\lambda\) customers enter a store in a single hour

\( X \sim \) number of customers enter the store in next hour

\(Pr(X = n) = \frac{\lambda^n e^{-\lambda}}{n!}~~~~~~\)

\( X \sim \) number of customers enter the store in next \(t\) hours

\(Pr(X = n) = \frac{(\lambda t)^n e^{-\lambda t}}{n!}~~~~~~\)

how long we will typically have to wait for the next customer to arrive?

if t increases the probability of Pr(T > t) decrease exponentially

a reaction involving those particles taking place at some average rate

how long should we expect to wait before this reaction occurs somewhere in the environment?

customers : instances of a chemical reaction
\(\lambda\) : the rate r at which the reaction occurs

we have n reactions with rates \(r_1, r_2, \dots, r_n\)

we can generate random time from exponential distribution

select one with probability \(P_i = \frac{r_i}{r_1 + r_2 + \dots + r_n} \)

a chemical reaction that attaches a phosphoryl group (\(PO_{3}^{-}\)) to an organic molecule.

Phosphoryl modifications serve as an information exchange

a sequence of phosphorylation and dephosphorylation events

transmit information within the cell about the amount of ligand binding being detected on the cell’s exterior.

methyl-accepting chemotaxis proteins (MCPs)

The pathway also includes a number of additional proteins, which all start with the prefix Che (short for “chemotaxis”).

in absence of MCP-ligand, this complex is more stable

phosphorylated CheA pass on its phosphoryl group to CheY

phosphorylated CheY interact with flagellar motor switch

change of flagellar rotation from counter-clockwise to clockwise

which in the absence of an increase in attractant occurs every 1 to 1.5 seconds

in presence of MCP-ligand, this complex is unstable

CheA is less able to autophosphorylate

We Can add this reactions to Gillespie algorithm

regardless of the absolute value of the concentration

cell returns to the same background tumbling frequency

Bacteria remember past concentrations using methylation

a methyl group (\(−CH_3\)) is added to an organic molecule

Every MCP receptor contains four methylation sites

MCPs, CheW, and CheA molecules form an array structure

Methylation reduces the negative charge on the receptors

When bound to MCPs, CheR methylates ligand-bound MCPs faster

the rate of MCP methylation by CheR is higher if the MCP is bound to a ligand

combinatorial explosion : means that building realistic models of biochemical systems at scale can be daunting.

rule-based modeling : a paradigm in which a potentially enormous number of reactions are specified by a much smaller collection of “rules” from which all reactions can be inferred.

Bacterial tumbling is robust to large sudden changes in attractant

in each state it returns to equilibrium quick and in same time

Imagine a glucose cube in an aqueous solution. As the cube dissolves, a gradient will form, with a glucose concentration that decreases outward from the cube.

The Beauty of E. coli’s Random Exploration Algorithm