9.4. Cooperativity[[Student version, January 17, 2003]] 317
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force, pN(relative extesion)(elastic rod model)Figure 9.5:(Experimental data with fit.) Linear plot of the stretching of DNA in regime C of Figure 9.3. A DNA
molecule with 38 800 basepairs was stretched with optical tweezers, in a buffer solution with pH 8.0. For each value
of the force, the ratio of the observed relative extension and the prediction of the inextensible elastic rod model is
plotted. The fact that this ratio is a linear function of applied force implies that the molecule has a simple elastic
stretching response to the applied force. The solid line is a straight line through the point (0pN,1), with fitted slope
1 /(BkBTr)=1/(1400pN). [Experimental points from Wang et al., 1997; data kindly supplied by M. D. Wang. ]
were first fit to the inextensible rod model as in Figure 9.4. Next, all of the extension data were
divided by the corresponding points obtained by extrapolating the inextensible rod model to higher
forces (corresponding to regime C of Figure 9.3). According to the previous paragraph, this residual
extension should be a linear function off—and the graph confirms this prediction. The slope lets
us read off the value of the stretch stiffness asBkBTr≈ 1400 pNfor DNA under the conditions of
this particular experiment.
9.4.3 Cooperativity in higher-dimensional systems gives rise to infinitely
sharp phase transitions
Equation 9.21 shows that the force-induced straightening transition becomes very sharp (the ef-
fective spring constantkbecomes small) whenγis big. That is, cooperativity, a local interaction
between neighbors in a chain, increases the sharpness of a global transition.
Actually, we are already familiar with cooperative transitions in our everyday, three-dimensional,
life. Suppose we take a beaker of water, carefully maintain it at a fixed, uniform temperature, and
allow it to come to equilibrium. Then the water will either be all liquid or all solid ice, depending
on whether the temperature is greater or less than 0◦C.This sharp transition can again be regarded
as a consequence of cooperativity. The interface between liquid water and ice has a surface tension,
afree-energy cost for introducing aboundarybetween these two phases, just as the parameter 2γ
in the polymer-straightening transition is the cost to create a boundary between a forward-directed
domain and one pointing against the applied force. This cost disfavors a mixed water/ice state,
making the water–ice transition discontinuous (infinitely sharp).
In contrast to the water/ice system, you found in Your Turn 9h(b) that the straightening of the
one-dimensional FJC by applied tension is never discontinuous, no matter how largeγmay be. We
say that the freezing of water is a truephase transition,but that such transitions are impossible
in one-dimensional systems with local interactions.
Wecan understand qualitatively why the physics of a cooperative, one-dimensional chain is so
different from analogous systems in three dimensions. Suppose the temperature in your glass of