Macromolecules : Binding Affinity and Fractional Occupancy. Regulation of the rates of reactions catalyzed by the macromolecule¶

Reaction `A <-> B` catalyzed when ligand `L` binds to `site 1` of macromolecule `M1`¶

IN-PROGRESS¶

In Part 1 we consider the un-catalyzed reaction A <-> B
In Part 2 we'll see what happens when catalysis is added

LAST REVISED: Dec. 3, 2023

In [1]:

import set_path      # Importing this module will add the project's home directory to sys.path

Added 'D:\Docs\- MY CODE\BioSimulations\life123-Win7' to sys.path

In [2]:

from src.modules.chemicals.chem_data import ChemData
from src.modules.reactions.reaction_dynamics import ReactionDynamics

import plotly.express as px

In [ ]:

PART 1 - Consider just the un-catalyzed reaction `A <-> B` by itself¶

In [3]:

# Initialize the system
chem1 = ChemData(names=["A", "B"])

# Reaction A <-> B , without catalysis (slow forward rate, relative to what we'll see in Part 2)
chem1.add_reaction(reactants="A", products="B",
                  forward_rate=1. , delta_G= -5000)

chem1.describe_reactions()

Number of reactions: 1 (at temp. 25 C)
0: A <-> B  (kF = 1 / kR = 0.13306 / delta_G = -5,000 / K = 7.5156) | 1st order in all reactants & products
Set of chemicals involved in the above reactions: {'B', 'A'}

In [4]:

# Set the initial concentrations of all the chemicals
dynamics1 = ReactionDynamics(chem_data=chem1)

dynamics1.set_conc(conc={"A": 100., "B": 20.},
                  snapshot=True)

dynamics1.describe_state()

SYSTEM STATE at Time t = 0:
2 species:
  Species 0 (A). Conc: 100.0
  Species 1 (B). Conc: 20.0
Set of chemicals involved in reactions: {'B', 'A'}

In [5]:

# Take the system to equilibrium
dynamics1.set_diagnostics()       # To save diagnostic information about the call to single_compartment_react()

# All of these settings are currently close to the default values... but subject to change; set for repeatability
dynamics1.set_thresholds(norm="norm_A", low=0.6, high=1.0, abort=1.44)
dynamics1.set_thresholds(norm="norm_B", low=0.04, high=0.6, abort=1.5)
dynamics1.set_step_factors(upshift=1.2, downshift=0.7, abort=0.4)
dynamics1.set_error_step_factor(0.3)

dynamics1.single_compartment_react(initial_step=0.02, reaction_duration=4.0,
                                  variable_steps=True, explain_variable_steps=False)

* INFO: the tentative time step (0.02) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.008) [Step started at t=0, and will rewind there]
80 total step(s) taken

In [6]:

#dynamics1.explain_time_advance()

In [7]:

dynamics1.plot_history(colors=['darkorange', 'green'], show_intervals=True, title_prefix="WITHOUT catalysis")

In [8]:

# Verify that the reaction has reached equilibrium
dynamics1.is_in_equilibrium(tolerance=5)

0: A <-> B
Final concentrations: [A] = 14.63 ; [B] = 105.4
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 7.20163
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 7.515573071036746
Discrepancy between the two values: 4.177 %
Reaction IS in equilibrium (within 5% tolerance)

Out[8]:

True

In [ ]:

PART 2¶

In [9]:

# Initialize the system
chem2 = ChemData(names=["A", "B", "L"])

chem2.add_macromolecules("M1")

chem2.set_binding_site_affinity("M1", site_number=1, ligand="L", Kd=5.0)

In [10]:

chem2.show_binding_affinities()        # Review the values we have given for the dissociation constants

M1  :
   Site 1 - Kd (dissociation const) for L : 5.0

In [ ]:

Define 2 reactions, one with and one without catalysis¶

In [11]:

# Reaction A <-> B , without catalysis (same as he had in part 1)
chem2.add_reaction(reactants="A", products="B",
                  forward_rate=1. , delta_G= -5000)

# Reaction A <-> B , WITH catalysis (fast forward rate)
#rxn = chem2.add_reaction(reactants="A", products="B",
#                        forward_rate=10. , delta_G= -5000)

#rxn.set_macro_enzyme(macromolecule="M1", site_number=1)

chem2.describe_reactions()

Number of reactions: 1 (at temp. 25 C)
0: A <-> B  (kF = 1 / kR = 0.13306 / delta_G = -5,000 / K = 7.5156) | 1st order in all reactants & products
Set of chemicals involved in the above reactions: {'B', 'A'}

Notice the thermodynamics is (as it should be!) the same - and hence the same equilibrium constants - but the kinetics are very different¶

In [ ]:

Start with no macromolecule ligand `L` present; hence, only the slow reaction is in effect¶

In [12]:

# Set the initial concentrations of all the chemicals, including the macromolecule

dynamics2 = ReactionDynamics(chem_data=chem2)
dynamics2.set_conc(conc={"A": 100., "B": 20., "L": 0.},
                  snapshot=True)      # The macromolecule ligand L is absent

dynamics2.set_macromolecules()      # By default, set counts to 1 for all the registered macromolecules

dynamics2.describe_state()

SYSTEM STATE at Time t = 0:
3 species:
  Species 0 (A). Conc: 100.0
  Species 1 (B). Conc: 20.0
  Species 2 (L). Conc: 0.0
Macro-molecules, with their counts:  {'M1': 1}
Fractional Occupancy at the various binding sites for each macro-molecule:
     M1 || 1: 0.0 (L)
Set of chemicals involved in reactions: {'B', 'A'}

In [ ]:

Take the initial system to equilibrium¶

In [13]:

dynamics2.set_diagnostics()       # To save diagnostic information about the call to single_compartment_react()

# All of these settings are currently close to the default values... but subject to change; set for repeatability
dynamics2.set_thresholds(norm="norm_A", low=0.6, high=1.0, abort=1.44)
dynamics2.set_thresholds(norm="norm_B", low=0.04, high=0.6, abort=1.5)
dynamics2.set_step_factors(upshift=1.2, downshift=0.7, abort=0.4)
dynamics2.set_error_step_factor(0.3)

dynamics2.single_compartment_react(initial_step=0.02, reaction_duration=4.0,
                                  variable_steps=True, explain_variable_steps=False)

* INFO: the tentative time step (0.02) leads to a least one norm value > its ABORT threshold:
      -> will backtrack, and re-do step with a SMALLER delta time, multiplied by 0.4 (set to 0.008) [Step started at t=0, and will rewind there]
80 total step(s) taken

In [14]:

dynamics2.explain_time_advance()

From time 0 to 0.008, in 1 step of 0.008
From time 0.008 to 0.0464, in 4 steps of 0.0096
From time 0.0464 to 0.08096, in 3 steps of 0.0115
From time 0.08096 to 0.2054, in 9 steps of 0.0138
From time 0.2054 to 0.3713, in 10 steps of 0.0166
From time 0.3713 to 0.5305, in 8 steps of 0.0199
From time 0.5305 to 0.6977, in 7 steps of 0.0239
From time 0.6977 to 0.8697, in 6 steps of 0.0287
From time 0.8697 to 1.042, in 5 steps of 0.0344
From time 1.042 to 1.207, in 4 steps of 0.0413
From time 1.207 to 1.355, in 3 steps of 0.0495
From time 1.355 to 1.534, in 3 steps of 0.0594
From time 1.534 to 1.676, in 2 steps of 0.0713
From time 1.676 to 2.019, in 4 steps of 0.0856
From time 2.019 to 2.224, in 2 steps of 0.103
From time 2.224 to 2.471, in 2 steps of 0.123
From time 2.471 to 2.767, in 2 steps of 0.148
From time 2.767 to 2.944, in 1 step of 0.177
From time 2.944 to 3.157, in 1 step of 0.213
From time 3.157 to 3.413, in 1 step of 0.256
From time 3.413 to 3.719, in 1 step of 0.307
From time 3.719 to 4.087, in 1 step of 0.368
(80 steps total)

In [15]:

dynamics2.plot_history(colors=['darkorange', 'green', 'darkblue'], show_intervals=True, title_prefix="WITHOUT catalysis")

In [16]:

# Verify that the reaction has reached equilibrium
dynamics2.is_in_equilibrium(tolerance=5)

0: A <-> B
Final concentrations: [A] = 14.63 ; [B] = 105.4
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 7.20163
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 7.515573071036746
Discrepancy between the two values: 4.177 %
Reaction IS in equilibrium (within 5% tolerance)

Out[16]:

True

In [ ]:

Macromolecules : Binding Affinity and Fractional Occupancy. Regulation of the rates of reactions catalyzed by the macromolecule¶

Reaction A <-> B catalyzed when ligand L binds to site 1 of macromolecule M1¶

IN-PROGRESS¶

PART 1 - Consider just the un-catalyzed reaction A <-> B by itself¶

PART 2¶

Define 2 reactions, one with and one without catalysis¶

Notice the thermodynamics is (as it should be!) the same - and hence the same equilibrium constants - but the kinetics are very different¶

Start with no macromolecule ligand L present; hence, only the slow reaction is in effect¶

Take the initial system to equilibrium¶

Reaction `A <-> B` catalyzed when ligand `L` binds to `site 1` of macromolecule `M1`¶

PART 1 - Consider just the un-catalyzed reaction `A <-> B` by itself¶

Start with no macromolecule ligand `L` present; hence, only the slow reaction is in effect¶