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: June 23, 2024 (using v. 1.0 beta36)

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 life123 import ChemData
from life123 import UniformCompartment

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 = UniformCompartment(chem_data=chem1, preset="fast")

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, error=0.3)

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

single_compartment_react(): the argument `reaction_duration` is deprecated; use `duration` instead
63 total step(s) taken
Number of step re-do's because of negative concentrations: 0
Number of step re-do's because of elective soft aborts: 1
Norm usage: {'norm_A': 31, 'norm_B': 18, 'norm_C': 18, 'norm_D': 18}

In [6]:

#dynamics1.explain_time_advance()

In [6]:

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

In [8]:

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

0: A <-> B
Final concentrations: [A] = 14.14 ; [B] = 105.9
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 7.48607
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 7.51557
Discrepancy between the two values: 0.3926 %
Reaction IS in equilibrium (within 1% 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 = UniformCompartment(chem_data=chem2, preset="fast")
dynamics2.set_conc(conc={"A": 100., "B": 20.},
                  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, error=0.3)

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

single_compartment_react(): the argument `reaction_duration` is deprecated; use `duration` instead
63 total step(s) taken
Number of step re-do's because of negative concentrations: 0
Number of step re-do's because of elective soft aborts: 1
Norm usage: {'norm_A': 31, 'norm_B': 18, 'norm_C': 18, 'norm_D': 18}

In [14]:

dynamics2.explain_time_advance()

From time 0 to 0.012, in 1 step of 0.012
From time 0.012 to 0.03, in 1 step of 0.018
From time 0.03 to 0.1164, in 6 steps of 0.0144
From time 0.1164 to 0.138, in 1 step of 0.0216
From time 0.138 to 0.2762, in 8 steps of 0.0173
From time 0.2762 to 0.3022, in 1 step of 0.0259
From time 0.3022 to 0.4473, in 7 steps of 0.0207
From time 0.4473 to 0.4784, in 1 step of 0.0311
From time 0.4784 to 0.6028, in 5 steps of 0.0249
From time 0.6028 to 0.6402, in 1 step of 0.0373
From time 0.6402 to 0.7596, in 4 steps of 0.0299
From time 0.7596 to 0.8044, in 1 step of 0.0448
From time 0.8044 to 0.9477, in 4 steps of 0.0358
From time 0.9477 to 1.001, in 1 step of 0.0537
From time 1.001 to 1.087, in 2 steps of 0.043
From time 1.087 to 1.152, in 1 step of 0.0645
From time 1.152 to 1.255, in 2 steps of 0.0516
From time 1.255 to 1.333, in 1 step of 0.0774
From time 1.333 to 1.456, in 2 steps of 0.0619
From time 1.456 to 1.549, in 1 step of 0.0929
From time 1.549 to 1.624, in 1 step of 0.0743
From time 1.624 to 1.735, in 1 step of 0.111
From time 1.735 to 1.824, in 1 step of 0.0892
From time 1.824 to 1.958, in 1 step of 0.134
From time 1.958 to 2.065, in 1 step of 0.107
From time 2.065 to 2.386, in 2 steps of 0.16
From time 2.386 to 2.867, in 2 steps of 0.241
From time 2.867 to 3.228, in 1 step of 0.361
From time 3.228 to 3.77, in 1 step of 0.542
From time 3.77 to 4.583, in 1 step of 0.812
(63 steps total)

In [15]:

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

In [17]:

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

0: A <-> B
Final concentrations: [A] = 14.14 ; [B] = 105.9
1. Ratio of reactant/product concentrations, adjusted for reaction orders: 7.48607
    Formula used:  [B] / [A]
2. Ratio of forward/reverse reaction rates: 7.51557
Discrepancy between the two values: 0.3926 %
Reaction IS in equilibrium (within 1% tolerance)

Out[17]:

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¶