-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathsteady.html
134 lines (123 loc) · 5.59 KB
/
steady.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
<!DOCTYPE html>
<html>
<head>
<title>Steady State</title>
<style type="text/css">
body {
background-color: #FFFFFF;
font-family: Verdana, sans-serif;
font-size: 12 px
}
</style>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\(','\\)']]}});
</script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
TeX: {
equationNumbers: { autoNumber: "AMS" },
extensions: ["AMSmath.js", "AMSsymbols.js"]
}
});
</script>
<script type="text/javascript"
src="https://c328740.ssl.cf1.rackcdn.com/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<script type="text/javascript" src="functions.js"></script>
</head>
<body>
<div style="display:none">
$
\newcommand{\conc}[1]{[\mathrm{#1}]}
\newcommand{\kcat}{k_{\mathrm{cat}}}
\newcommand{\kmmon}{\kon^{\mathrm{ES}}}
\newcommand{\kmmoff}{\koff^{\mathrm{ES}}}
\newcommand{\koff}{k_{\mathrm{off}}}
\newcommand{\kon}{k_{\mathrm{on}}}
\newcommand{\ss}{\mathrm{SS}}
$
</div>
<h2>
The Steady State: A Key Description of Biology
</h2>
<p style="text-align:center"><img src="images/waterfall-steady-state.gif" alt="Desktop waterfall - a steady state" /></p>
<h3>
Background
</h3>
<ul>
<li> A steady state is characterized by unchanging probabilities/concentrations, but matter or probability may be flowing. </li>
<li> Mathematically, all time derivatives are zero. </li>
<li> Although the cell is not strictly in a steady state, many processes can be modeled reasonably as steady: think "homeostasis". </li>
<li> <a href="javascript:changeTo('steady','equil')">Equilibrium</a> is a special steady state in which no net flows of material or probability occur. </li>
<li> Steady states with flows (i.e., those out of equilibrium) require input of matter or energy. They are not self-sustaining. The "desktop waterfall" shown above must be plugged in for the flow to be maintained. </li>
<li> Steady states with flows typically are amenable to a simple mathematical treatment. They also are convenient modules for connecting to other parts of a larger system - the sources and sinks of flows.
</ul>
<h3> Schematically </h3>
<p> A steady state consists of one or more inputs and one or more outputs, with each component unchanging in time. </p>
<p style="text-align:center"><img src="images/steady-state-schematic-simple.gif" alt="Simple schematic of a steady state" /></p>
\begin{equation}
\label{ss-schematic}
\frac{ d \conc{A} }{ dt } = 0
\hspace{1cm}
\frac{ d \conc{B} }{ dt } = 0
\end{equation}
<p> A more typical (and complex) case includes multiple inputs/outputs and an internal cycle </p>
<p style="text-align:center"><img src="images/steady-state-schematic-cycle.gif" alt="Complex schematic of a steady state" /></p>
<h3>
Key Biological Examples
</h3>
<ul>
<li> Michaelis-Menten catalytic cycle </li>
<li> Citric acid cycle </li>
<li> Molecular locomotion </li>
<li> Active <a href="javascript:changeTo('steady','transport')">transport</a> </li>
</ul>
<h3>
Steady-state analysis of a Michaelis-Menten (MM) process
</h3>
<p>
A standard MM process models conversion of a substrate (S) to a product (P), <a href="javascript:changeTo('steady','catalysis')">catalyzed</a> by an enzyme (E) after formation of a bound-but-uncatalyzed complex (ES).
</p>
<p style="text-align:center"><img src="images/mm-simple-equation.gif" alt="Michaelis-Menten simple equation" /></p>
<p>
The simple MM model can also be viewed as a cycle because the enzyme E is re-used. Blue arrows indicate steady net flows.
</p>
<p style="text-align:center"><img src="images/mm-simple-cycle.gif" alt="Michaelis-Menten simple cycle" /></p>
(The standard MM process here can be contrasted with the <a href="javascript:changeTo('steady','equil')">corrected MM cycle</a> that allows for reverse events and physical single-step processes.)
<p>
A steady state will occur if P is removed at the same rate as S is added.
Mathematically, for steady state, we set the time derivative of the ES complex to zero.
\begin{equation}
\label{michaelis}
\frac{d\conc{ES}}{dt}
= \conc{E}\conc{S} \,\kmmon
- \conc{ES} \, \kmmoff
- \conc{ES} \, \kcat
= 0
\end{equation}
</p>
<p> The result yields what looks like a <a href="javascript:changeTo('steady','binding')">dissociation constant</a> in terms of the steady-state (SS) concentrations: </p>
\begin{equation}
\label{km}
\frac{ \conc{E}^{\ss} \conc{S}^{\ss} }{ \conc{ES}^{\ss} }
= \frac{ \kmmoff + \kcat }{ \kmmon }
\equiv K_M
\end{equation}
<p> In words, in the steady state, the ratio of concentrations on the left assumes the constant value given by the particular ratio of rate constants in the middle.
The effective "equilibrium" constant $K_M$ is conventionally defined but not strictly needed.
</p>
<p> The basic steady state result \eqref{km} can be used to calculate other quantities of interest, such as the overall rate of product production </p>
\begin{equation}
\label{mm-product}
\kcat \conc{ES}^{\ss} = \conc{E}^{\ss} \conc{S}^{\ss} \frac{\kcat}{K_M}
\end{equation}
now given in terms of the steady-state E and S concentrations, which should be known.
<h3>
The standard MM model is unphysical
</h3>
<p>
All molecular processes are reversible, so any model with a uni-directional arrow is necesarily approximate: see the discussion of <a href="javascript:changeTo('steady','cycles')">cycles</a>.
The full MM cycle, allowing for reverse events and permitting only single-step processes, is subjected to a (more complicated) steady-state analysis in an advanced section.
</p>
</body>
</html>