Equilibrium formalism

Equilibrium formalism#

In preparation for our discussion of conformational coupling in Receptor Dimers, this section illustrates the equilibrium formalism for a monomeric receptor model. In essence, we will repeat our analysis of sequential Ligand Binding and Equilibrium Binding Curves, but we will do so using a helpful notation, developed in [CS20] (see Welcome to Receptors).

Two equilibrium constants in the three-state receptor model#

The receptor model’s state-transition diagram has the topology of a symmetric directed path graph on 3 vertices. The weighted rooted spanning tree that specifies the parameters of equilibrium receptor model is shown below.

var('a b c kappa_b kappa_c')
T = DiGraph([[a,b,c],[(b,a),(c,b)]])
T.set_edge_label(b,a,kappa_b)
T.set_edge_label(c,b,kappa_c)
T.plot(figsize=6,pos={a:(0,0),b:(2,0),c:(4,0)},edge_labels=True,graph_border=True,vertex_size=1000)

A module that was compiled using NumPy 1.x cannot be run in
NumPy 2.2.6 as it may crash. To support both 1.x and 2.x
versions of NumPy, modules must be compiled with NumPy 2.0.
Some module may need to rebuild instead e.g. with 'pybind11>=2.12'.

If you are a user of the module, the easiest solution will be to
downgrade to 'numpy<2' or try to upgrade the affected module.
We expect that some modules will need time to support NumPy 2.

Traceback (most recent call last):  File "/usr/lib/python3.10/runpy.py", line 196, in _run_module_as_main
    return _run_code(code, main_globals, None,
  File "/usr/lib/python3.10/runpy.py", line 86, in _run_code
    exec(code, run_globals)
  File "/usr/lib/python3/dist-packages/sage/repl/ipython_kernel/__main__.py", line 3, in <module>
    IPKernelApp.launch_instance(kernel_class=SageKernel)
  File "/usr/lib/python3/dist-packages/traitlets/config/application.py", line 846, in launch_instance
    app.start()
  File "/usr/lib/python3/dist-packages/ipykernel/kernelapp.py", line 677, in start
    self.io_loop.start()
  File "/usr/lib/python3/dist-packages/tornado/platform/asyncio.py", line 199, in start
    self.asyncio_loop.run_forever()
  File "/usr/lib/python3.10/asyncio/base_events.py", line 603, in run_forever
    self._run_once()
  File "/usr/lib/python3.10/asyncio/base_events.py", line 1909, in _run_once
    handle._run()
  File "/usr/lib/python3.10/asyncio/events.py", line 80, in _run
    self._context.run(self._callback, *self._args)
  File "/usr/lib/python3/dist-packages/ipykernel/kernelbase.py", line 461, in dispatch_queue
    await self.process_one()
  File "/usr/lib/python3/dist-packages/ipykernel/kernelbase.py", line 450, in process_one
    await dispatch(*args)
  File "/usr/lib/python3/dist-packages/ipykernel/kernelbase.py", line 357, in dispatch_shell
    await result
  File "/usr/lib/python3/dist-packages/ipykernel/kernelbase.py", line 652, in execute_request
    reply_content = await reply_content
  File "/usr/lib/python3/dist-packages/ipykernel/ipkernel.py", line 353, in do_execute
    res = shell.run_cell(code, store_history=store_history, silent=silent)
  File "/usr/lib/python3/dist-packages/ipykernel/zmqshell.py", line 532, in run_cell
    return super().run_cell(*args, **kwargs)
  File "/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py", line 2914, in run_cell
    result = self._run_cell(
  File "/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py", line 2960, in _run_cell
    return runner(coro)
  File "/usr/lib/python3/dist-packages/IPython/core/async_helpers.py", line 78, in _pseudo_sync_runner
    coro.send(None)
  File "/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py", line 3185, in run_cell_async
    has_raised = await self.run_ast_nodes(code_ast.body, cell_name,
  File "/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py", line 3377, in run_ast_nodes
    if (await self.run_code(code, result,  async_=asy)):
  File "/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py", line 3457, in run_code
    exec(code_obj, self.user_global_ns, self.user_ns)
  File "/tmp/ipykernel_11313/3221940383.py", line 5, in <module>
    T.plot(figsize=Integer(6),pos={a:(Integer(0),Integer(0)),b:(Integer(2),Integer(0)),c:(Integer(4),Integer(0))},edge_labels=True,graph_border=True,vertex_size=Integer(1000))
  File "/usr/lib/python3/dist-packages/IPython/core/displayhook.py", line 262, in __call__
    format_dict, md_dict = self.compute_format_data(result)
  File "/usr/lib/python3/dist-packages/IPython/core/displayhook.py", line 151, in compute_format_data
    return self.shell.display_formatter.format(result)
  File "/usr/lib/python3/dist-packages/sage/repl/display/formatter.py", line 181, in format
    sage_format, sage_metadata = self.dm.displayhook(obj)
  File "/usr/lib/python3/dist-packages/sage/repl/rich_output/display_manager.py", line 825, in displayhook
    plain_text, rich_output = self._rich_output_formatter(obj, dict())
  File "/usr/lib/python3/dist-packages/sage/repl/rich_output/display_manager.py", line 643, in _rich_output_formatter
    rich_output = self._call_rich_repr(obj, rich_repr_kwds)
  File "/usr/lib/python3/dist-packages/sage/repl/rich_output/display_manager.py", line 603, in _call_rich_repr
    return obj._rich_repr_(self)
  File "/usr/lib/python3/dist-packages/sage/plot/graphics.py", line 1000, in _rich_repr_
    return display_manager.graphics_from_save(
  File "/usr/lib/python3/dist-packages/sage/repl/rich_output/display_manager.py", line 731, in graphics_from_save
    save_function(filename, **kwds)
  File "/usr/lib/python3/dist-packages/sage/misc/decorators.py", line 410, in wrapper
    return func(*args, **kwds)
  File "/usr/lib/python3/dist-packages/sage/plot/graphics.py", line 3296, in save
    from matplotlib import rcParams
  File "/usr/lib/python3/dist-packages/matplotlib/__init__.py", line 109, in <module>
    from . import _api, _version, cbook, docstring, rcsetup
  File "/usr/lib/python3/dist-packages/matplotlib/rcsetup.py", line 27, in <module>
    from matplotlib.colors import Colormap, is_color_like
  File "/usr/lib/python3/dist-packages/matplotlib/colors.py", line 56, in <module>
    from matplotlib import _api, cbook, scale
  File "/usr/lib/python3/dist-packages/matplotlib/scale.py", line 23, in <module>
    from matplotlib.ticker import (
  File "/usr/lib/python3/dist-packages/matplotlib/ticker.py", line 136, in <module>
    from matplotlib import transforms as mtransforms
  File "/usr/lib/python3/dist-packages/matplotlib/transforms.py", line 46, in <module>
    from matplotlib._path import (

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
AttributeError: _ARRAY_API not found

/usr/lib/python3/dist-packages/sage/repl/rich_output/display_manager.py:608: RichReprWarning: Exception in _rich_repr_ while displaying object: numpy.core.multiarray failed to import
  warnings.warn(

A module that was compiled using NumPy 1.x cannot be run in
NumPy 2.2.6 as it may crash. To support both 1.x and 2.x
versions of NumPy, modules must be compiled with NumPy 2.0.
Some module may need to rebuild instead e.g. with 'pybind11>=2.12'.

If you are a user of the module, the easiest solution will be to
downgrade to 'numpy<2' or try to upgrade the affected module.
We expect that some modules will need time to support NumPy 2.

Traceback (most recent call last):  File "/usr/lib/python3.10/runpy.py", line 196, in _run_module_as_main
    return _run_code(code, main_globals, None,
  File "/usr/lib/python3.10/runpy.py", line 86, in _run_code
    exec(code, run_globals)
  File "/usr/lib/python3/dist-packages/sage/repl/ipython_kernel/__main__.py", line 3, in <module>
    IPKernelApp.launch_instance(kernel_class=SageKernel)
  File "/usr/lib/python3/dist-packages/traitlets/config/application.py", line 846, in launch_instance
    app.start()
  File "/usr/lib/python3/dist-packages/ipykernel/kernelapp.py", line 677, in start
    self.io_loop.start()
  File "/usr/lib/python3/dist-packages/tornado/platform/asyncio.py", line 199, in start
    self.asyncio_loop.run_forever()
  File "/usr/lib/python3.10/asyncio/base_events.py", line 603, in run_forever
    self._run_once()
  File "/usr/lib/python3.10/asyncio/base_events.py", line 1909, in _run_once
    handle._run()
  File "/usr/lib/python3.10/asyncio/events.py", line 80, in _run
    self._context.run(self._callback, *self._args)
  File "/usr/lib/python3/dist-packages/ipykernel/kernelbase.py", line 461, in dispatch_queue
    await self.process_one()
  File "/usr/lib/python3/dist-packages/ipykernel/kernelbase.py", line 450, in process_one
    await dispatch(*args)
  File "/usr/lib/python3/dist-packages/ipykernel/kernelbase.py", line 357, in dispatch_shell
    await result
  File "/usr/lib/python3/dist-packages/ipykernel/kernelbase.py", line 652, in execute_request
    reply_content = await reply_content
  File "/usr/lib/python3/dist-packages/ipykernel/ipkernel.py", line 353, in do_execute
    res = shell.run_cell(code, store_history=store_history, silent=silent)
  File "/usr/lib/python3/dist-packages/ipykernel/zmqshell.py", line 532, in run_cell
    return super().run_cell(*args, **kwargs)
  File "/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py", line 2914, in run_cell
    result = self._run_cell(
  File "/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py", line 2960, in _run_cell
    return runner(coro)
  File "/usr/lib/python3/dist-packages/IPython/core/async_helpers.py", line 78, in _pseudo_sync_runner
    coro.send(None)
  File "/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py", line 3185, in run_cell_async
    has_raised = await self.run_ast_nodes(code_ast.body, cell_name,
  File "/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py", line 3377, in run_ast_nodes
    if (await self.run_code(code, result,  async_=asy)):
  File "/usr/lib/python3/dist-packages/IPython/core/interactiveshell.py", line 3457, in run_code
    exec(code_obj, self.user_global_ns, self.user_ns)
  File "/tmp/ipykernel_11313/3221940383.py", line 5, in <module>
    T.plot(figsize=Integer(6),pos={a:(Integer(0),Integer(0)),b:(Integer(2),Integer(0)),c:(Integer(4),Integer(0))},edge_labels=True,graph_border=True,vertex_size=Integer(1000))
  File "/usr/lib/python3/dist-packages/IPython/core/displayhook.py", line 262, in __call__
    format_dict, md_dict = self.compute_format_data(result)
  File "/usr/lib/python3/dist-packages/IPython/core/displayhook.py", line 151, in compute_format_data
    return self.shell.display_formatter.format(result)
  File "/usr/lib/python3/dist-packages/sage/repl/display/formatter.py", line 186, in format
    if (not isinstance(obj, (IPYTHON_NATIVE_TYPES, Figure)) and
  File "/usr/lib/python3/dist-packages/matplotlib/__init__.py", line 109, in <module>
    from . import _api, _version, cbook, docstring, rcsetup
  File "/usr/lib/python3/dist-packages/matplotlib/rcsetup.py", line 27, in <module>
    from matplotlib.colors import Colormap, is_color_like
  File "/usr/lib/python3/dist-packages/matplotlib/colors.py", line 56, in <module>
    from matplotlib import _api, cbook, scale
  File "/usr/lib/python3/dist-packages/matplotlib/scale.py", line 23, in <module>
    from matplotlib.ticker import (
  File "/usr/lib/python3/dist-packages/matplotlib/ticker.py", line 136, in <module>
    from matplotlib import transforms as mtransforms
  File "/usr/lib/python3/dist-packages/matplotlib/transforms.py", line 46, in <module>
    from matplotlib._path import (

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
AttributeError: _ARRAY_API not found

---------------------------------------------------------------------------
ImportError                               Traceback (most recent call last)
/tmp/ipykernel_11313/3221940383.py in <module>
      3 T.set_edge_label(b,a,kappa_b)
      4 T.set_edge_label(c,b,kappa_c)
----> 5 T.plot(figsize=Integer(6),pos={a:(Integer(0),Integer(0)),b:(Integer(2),Integer(0)),c:(Integer(4),Integer(0))},edge_labels=True,graph_border=True,vertex_size=Integer(1000))

/usr/lib/python3/dist-packages/IPython/core/displayhook.py in __call__(self, result)
    260             self.start_displayhook()
    261             self.write_output_prompt()
--> 262             format_dict, md_dict = self.compute_format_data(result)
    263             self.update_user_ns(result)
    264             self.fill_exec_result(result)

/usr/lib/python3/dist-packages/IPython/core/displayhook.py in compute_format_data(self, result)
    149 
    150         """
--> 151         return self.shell.display_formatter.format(result)
    152 
    153     # This can be set to True by the write_output_prompt method in a subclass

/usr/lib/python3/dist-packages/sage/repl/display/formatter.py in format(self, obj, include, exclude)
    184         # use Sage rich output for any except those native to IPython, but only
    185         # if it is not plain and dull
--> 186         if (not isinstance(obj, (IPYTHON_NATIVE_TYPES, Figure)) and
    187             not set(sage_format.keys()).issubset([PLAIN_TEXT])):
    188             return sage_format, sage_metadata

/usr/lib/python3/dist-packages/sage/misc/lazy_import.pyx in sage.misc.lazy_import.LazyImport.__instancecheck__ (build/cythonized/sage/misc/lazy_import.c:7695)()
    914             True
    915         """
--> 916         return isinstance(x, self.get_object())
    917 
    918     def __subclasscheck__(self, x):

/usr/lib/python3/dist-packages/sage/misc/lazy_import.pyx in sage.misc.lazy_import.LazyImport.get_object (build/cythonized/sage/misc/lazy_import.c:2612)()
    215         if likely(self._object is not None):
    216             return self._object
--> 217         return self._get_object()
    218 
    219     cpdef _get_object(self):

/usr/lib/python3/dist-packages/sage/misc/lazy_import.pyx in sage.misc.lazy_import.LazyImport._get_object (build/cythonized/sage/misc/lazy_import.c:3073)()
    255             if self._feature:
    256                 raise FeatureNotPresentError(self._feature, reason=f'Importing {self._name} failed: {e}')
--> 257             raise
    258 
    259         name = self._as_name

/usr/lib/python3/dist-packages/sage/misc/lazy_import.pyx in sage.misc.lazy_import.LazyImport._get_object (build/cythonized/sage/misc/lazy_import.c:2935)()
    251 
    252         try:
--> 253             self._object = getattr(__import__(self._module, {}, {}, [self._name]), self._name)
    254         except ImportError as e:
    255             if self._feature:

/usr/lib/python3/dist-packages/matplotlib/__init__.py in <module>
    107 # cbook must import matplotlib only within function
    108 # definitions, so it is safe to import from it here.
--> 109 from . import _api, _version, cbook, docstring, rcsetup
    110 from matplotlib.cbook import MatplotlibDeprecationWarning, sanitize_sequence
    111 from matplotlib.cbook import mplDeprecation  # deprecated

/usr/lib/python3/dist-packages/matplotlib/rcsetup.py in <module>
     25 from matplotlib import _api, cbook
     26 from matplotlib.cbook import ls_mapper
---> 27 from matplotlib.colors import Colormap, is_color_like
     28 from matplotlib.fontconfig_pattern import parse_fontconfig_pattern
     29 from matplotlib._enums import JoinStyle, CapStyle

/usr/lib/python3/dist-packages/matplotlib/colors.py in <module>
     54 import matplotlib as mpl
     55 import numpy as np
---> 56 from matplotlib import _api, cbook, scale
     57 from ._color_data import BASE_COLORS, TABLEAU_COLORS, CSS4_COLORS, XKCD_COLORS
     58 

/usr/lib/python3/dist-packages/matplotlib/scale.py in <module>
     21 import matplotlib as mpl
     22 from matplotlib import _api, docstring
---> 23 from matplotlib.ticker import (
     24     NullFormatter, ScalarFormatter, LogFormatterSciNotation, LogitFormatter,
     25     NullLocator, LogLocator, AutoLocator, AutoMinorLocator,

/usr/lib/python3/dist-packages/matplotlib/ticker.py in <module>
    134 import matplotlib as mpl
    135 from matplotlib import _api, cbook
--> 136 from matplotlib import transforms as mtransforms
    137 
    138 _log = logging.getLogger(__name__)

/usr/lib/python3/dist-packages/matplotlib/transforms.py in <module>
     44 
     45 from matplotlib import _api
---> 46 from matplotlib._path import (
     47     affine_transform, count_bboxes_overlapping_bbox, update_path_extents)
     48 from .path import Path

ImportError: numpy.core.multiarray failed to import

In this diagram, kappa_b and kappa_c are dimensionless equilibrium constants that will often be rendered as \(\kappab\) and \(\kappac\) in the mathematical expressions below.

The edges of the spanning tree are directed backwards, i.e., the forward reaction is against the direction of the arrow. For example, the reaction labelled with the equilibrium constant \(\kappab\) has \(a\) as reactant and \(b\) as product; consequently, increasing \(\kappab\) decreases the equilibrium probability (relative fraction) of state \(a\) and increases the probability of state \(b\).

The three states are labelled so that the reactant comes before the product in dictionary order (\(a\) to \(b\) to \(c\)). The subscript of the equilibrium constants \(\kappab\) and \(\kappac\) are chosen to match the label of the products.

Including the ligand concentration dependence of equilibrium constants#

The dependence of the two equilibrium constants on ligand concentration is incorporated by defining \(\kappab = \kappabstar x\) and \(\kappac = \kappacstar x\) where \(\kappabstar\) and \(\kappacstar\) are association constants with physical dimension of inverse concentration, and \(x\) is ligand concentration.

var('a b c x kb kc')
T = DiGraph([[a,b,c],[(b,a),(c,b)]])
T.set_edge_label(b,a,kb*x)
T.set_edge_label(c,b,kc*x)
T.plot(figsize=6,pos={a:(0,0),b:(3,0),c:(6,0)},edge_labels=True,graph_border=True,vertex_size=1000)

The probability of each receptor state#

For the receptor model above, the probability of state \(i\) is given by \(\pi_i = z_i / z_T\) where \(z_T= \textstyle \sum_i z_i\), and

(1)#\[z_a = 1 \quad z_b = \kappab = \kappabstar x \quad z_c =\kappab \kappac = \kappabstar \kappacstar x^2 \, .\]

That is,

(2)#\[\begin{equation} \pi_a = \frac{1}{1+ \kappabstar x + \kappabstar \kappacstar x^2} \, , \quad \pi_b = \frac{\kappabstar x}{1+ \kappabstar x + \kappabstar \kappacstar x^2} \quad \mbox{and} \quad \pi_c = \frac{\kappabstar \kappacstar x^2 }{1+ \kappabstar x + \kappabstar \kappacstar x^2 } \, . \end{equation}\]

It is helpful to present this set of rational functions using the following compact notation:

(3)#\[\begin{equation} [ \pi_a : \pi_b : \pi_c ] = [1 :\kappab : \kappab \kappac ] = [1 : \kappabstar x :\kappabstar \kappacstar x^2 ] \, . \end{equation}\]

In expressions of this kind, it is understood that

(4)#\[\begin{equation} [ z_1 \! : \! z_2 : \! \cdots \! : \! z_n ] = [ \lambda z_1 \! : \! \lambda z_2 : \! \cdots \! : \! \lambda z_n ] \end{equation}\]

for any \(\lambda \neq 0\). Furthermore, \(\lambda = 1/\sum_i x_n\) gives the probability distribution \(\pi = (\pi_1, \pi_2, \ldots, \pi_n)\) where \(1=\sum_i \pi_i\). Prior to normalization, we will refer to \([ z_1 \! : \! z_2 : \! \cdots \! : \! z_n ]\) as relative probabilities for each receptor state.

From spanning tree to relative probabilities#

Using a spanning tree as the specification for the receptor model, the following SageMath commands extract symbolic expressions for the fraction of receptors in each state.

paths = T.all_simple_paths(starting_vertices=[a,b,c],ending_vertices=[a],trivial=True)
print(paths)

The list paths has length 3. paths[0]=[a]. paths[1]=[b,a]. paths[2]=[c,b,a]. These are vertices encountered in paths beginning at vertex 0 (a), 1 (b), and 2 (c) and (in each case) ending at vertex a.

The relative probability of each state is obtained as the product of the edge weights in each path, with the trivial path yielding 1 (an empty product).

paths = T.all_simple_paths(starting_vertices=[a,b,c],ending_vertices=[a],trivial=True)
print(paths)
z = []
for p in paths:
    w = 1
    for i in range(len(p)-1):
       w = w*T.edge_label(p[i],p[i+1])
    z.append(w)
print(z)

The list z also has length 3. z[0]=1. z[1]=kb*x. z[2]=kb*kc*x^2. These are the relative probabilities \(z_a\), \(z_b\), and \(z_c\), which agree with our previous analytical calculation (1).

Symbolic expressions for the normalized probabilities are found as follows.

ztot = sum(z);
prob = []
for i in range(len(z)):
    prob.append(z[i]/ztot)
print(prob)

Equilibrium binding curve#

After choosing values for the association constants kb and kc, equilibrium binding curves can be plotted.

xmin=0.01; xmax=100;
params = {kb:1,kc:1}
p = [0]*3
col = ['red','green','blue']
for i in range(3):
    p[i] = plot_semilogx(prob[i].subs(params), (x, xmin, xmax), color=col[i], legend_label='p[%s]'%i, axes_labels=['x', 'probability'])
    print('p[%s] ='%i,prob[i].subs(params))
show(sum(p))

The above plot can be compared to those in the previous section (Ligand Binding). The only distinction is the usage of the notation that is the focus of this section. This notation is especially helpful in the analysis of conformational coupling of receptor dimers and higher-order oligomers.

References#

[CS20]

Gregory Douglas Conradi Smith. Allostery in oligomeric receptor models. Mathematical Medicine and Biology, 37:313–333, 2020.