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<!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/javascript" async
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equationNumbers: { autoNumber: "AMS" },
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</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" height = "200" hspace = "20" alt="Desktop waterfall - a steady state" >
<img src="images/steady-state-schematic-cycle.gif" height = "200" hspace = "20" alt="Complex schematic of a steady state" >
</p>
<p>
The steady state - when populations, concentrations and spatial distributions are unchanging in time - is one of the most important physical concepts for understanding cell biology.
This is <i>not</i> to say that cells are generally <i>in</i> steady states: after all, the <a href="https://en.wikipedia.org/wiki/Cell_cycle">cell cycle</a> is a never-ending repeated sequence of changes of from one stage of life to another.
In many cases, however, a steady state is a reasonable approximation to a short (enough) window of time in a cellular process, perhaps in a localized region of interest.
Think <a href="https://en.wikipedia.org/wiki/Homeostasis">"homeostasis"</a> (but beware of biologists' informal use of the word "equilibrium" which we clarify below).
Even if/when a steady state does not hold even approximately, it is still an essential conceptual reference point.
You absolutely must understand it.
</p>
<p>
Two examples of (potential) steady states are sketched above.
A table-top waterfall will be in a steady state as water continuously is pumped up from the lower reservoir - to which it returns by gravity.
A complex chemical cycle, meant to evoke the <a href="https://en.wikipedia.org/wiki/Citric_acid_cycle">citric acid cycle</a>, takes several types of molecules as inputs and catalytically changes them into different output molecules.
Note that both examples require the input of energy and/or matter.
And neither will be in a steady state if the inputs are removed - or in the transient period after the system is initiated.
Other examples, observed over suitable time windows, include motion by <a href="http://www.physicallensonthecell.org/molecular-machinery/molecular-motors">motor proteins</a> (which requires constant input of ATP) and <a href="http://www.physicallensonthecell.org/molecular-machinery/pumps-and-transporters">active transport</a> (which requires a driving electro/chemical gradient or ATP).
Below, we'll look more closely at a Michaelis-Menten process used to model catalytic and biosynthetic processes.
</p>
<p> Schematically, 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>
<p style="text-align:center"><a name="dummy"><img src="equations/steady.html_eq1.png" height="50" /></a> (1)</p>
<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>
<p>
Let's define a steady state more precisely.
We'll restrict ourselves to considering a dsicrete set of <a href="http://www.physicallensonthecell.org/chemical-physics/basics-states-kinetics">states</a> (e.g., chemical and conformational states, possibly of many different types of molecules) simply numbered $i = 1, 2, \ldots$ or $j = 1, 2, \ldots$ with concentrations $[i]$ or $[j]$ which interconvert according to <a href="http://www.physicallensonthecell.org/chemical-physics/basics-mass-action-kinetics">first-order rate constants</a> $k_{ij}$.
Then the steady-state condition that every concentration be unchanging in time will be satisfied if, and only if, the flow into a state (i.e., the number of molecules changing into that state) exactly balances the flow out of a state (conversions to other states):
<p style="text-align:center"><a name="ss"><img src="equations/steady.html_eq2.png" height="50" /></a> (2)</p>
Since the concentrations are steady (unchanging) in time, the time derivative of every derivative will be zero, as we explore further below.
Note that <i>spatial derivatives (gradients) need not vanish in a steady state:</i> in a (hypothetically perfectly steady) cell, a molecule could be produced in one region and consumed in another.
</p>
<h3>
Equilibrium is a special steady state
</h3>
<p>
What's the difference between a steady state and <a href="http://www.physicallensonthecell.org/chemical-physics/equilibrium-means-detailed-balance">equilibrium</a>?
That's a little bit of a trick question because equilibrium <i>is</i> a steady state!
Equilibrium is a very special steady state, however, in which the condition <a href="#ss">(2)</a> is satisfied in a special way - namely, by the <i>stricter</i> condition of detailed balance,
<p style="text-align:center"><a name="db"><img src="equations/steady.html_eq3.png" height="50" /></a> (3)</p>
That is, the number of transitions per second from $i$ to $j$ is exactly balanced by reverse transitions.
Because this holds for <i>every</i> pair of states in equilibrium, it is said to hold "in detail."
Intuitively, it should be clear that if state $i$ experiences equal in and out flow with every other state, then its population/concentration cannot change, and so condition <a href="#ss">(2)</a> will be satisfied.
</p>
<h3>
Non-equilibrium steady states require inputs and outputs
</h3>
In the sketch of the waterfall at the top, I was careful to include the power cord because non-equilibrium steady states (when there are net flows through a system) require input of energy implicitly or explicitly.
<i>Steady states are not self-sustaining.</i>
Note that the input and removal of matter from a system implies the use of energy to enact those processes.
Without the input of energy (and/or material) a system will relax to equilibrium, and biophysicists sometimes like to say, "Equilibrium equals death."
Equilibrium's condition of detailed balance <a href="#db">(3)</a>, which also holds for flow in real space, means there is no net flow of matter or chemical processes, which are absolutely required for life.
What a cell does, after all, is to orchestrate a complex series of <i>directed</i> processes: signals travel from cell surface to nucleus; genes are transcribed and translated.
If these processes don't overwhelmingly proceed in a single direction, the cell just won't work.
In fact, if you think about it, a lot the cell's use of <a href="http://www.physicallensonthecell.org/energy-economy/activated-carriers">activated carriers</a> like <a href="http://www.physicallensonthecell.org/energy-economy/atp-great-carrier">ATP</a> goes toward maintaining the directionality of signaling processes - e.g., via <a href="http://www.physicallensonthecell.org/molecular-machinery/phosphorylation-methylation-etc">phosphorylation</a> - rather than doing work per se.
<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="http://www.physicallensonthecell.org/physical-molecular-processes/chemical-reactions-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="http://www.physicallensonthecell.org/chemical-physics/equilibrium-means-detailed-balance">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.
<p style="text-align:center"><a name="michaelis"><img src="equations/steady.html_eq4.png" height="50" /></a> (4)</p>
</p>
<p> The result yields what looks like a <a href="http://www.physicallensonthecell.org/physical-molecular-processes/binding-kinetics-thermodynamics">dissociation constant</a> in terms of the steady-state (SS) concentrations: </p>
<p style="text-align:center"><a name="km"><img src="equations/steady.html_eq5.png" height="50" /></a> (5)</p>
<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 <a href="#km">(5)</a> can be used to calculate other quantities of interest, such as the overall rate of product production </p>
<p style="text-align:center"><a name="dummy"><img src="equations/steady.html_eq6.png" height="50" /></a> (6)</p>
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="http://www.physicallensonthecell.org/basics-mass-action-kinetics/cycles-and-constraints">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>
<h3>
Exercises
</h3>
<ol>
<li> Show that the detailed balance condition <a href="#db">(3)</a> leads to the steady-state condition <a href="#ss">(2)</a>.</li>
<li> Construct an example of a <i>non-equilibrium</i> steady state consisting of three states ($i = 1, 2, 3$) - i.e., which satistfies Eq. <a href="#ss">(2)</a> but <i>not</i> <a href="#db">(3)</a>. Hint: Try starting from a very symmetric detailed balance system and adjusting one rate constant at a time.</li>
</ol>
<h3>
References
</h3>
<ul>
<li> Wikipedia's <a href="http://en.wikipedia.org/wiki/Steady_state_%28chemistry%29">Steady state (chemistry)</a> discussion is very accessible and useful. </li>
<li> D. M. Zuckerman, "Statistical Physics of Biomolecules: An Introduction" (CRC Press) discusses steady states generally, as well as analyzing the simple Michaelis-Menten model. </li>
<li> J. M. Berg et al., "Biochemistry" (W. H. Freeman). <a href="http://www.ncbi.nlm.nih.gov/books/NBK22430/">The 2002 edition is online for free</a>. </li>
<li> B. Alberts et al., "Molecular Biology of the Cell" (Garland Science). </li>
</ul>
</body>
</html>