Intramolecular dynamics of linear macromolecules by fluorescence correlation spectroscopy

A theoretical description of the dynamics of DNA molecules and actin filaments in solution as measured experimentally by fluorescence correlation spectroscopy is provided and compared to recent experimental results. Particular attention is paid to the contribution of the intramolecular dynamics to the fluorescence correlation function. Using a semiflexible chain model, a theoretical expression is presented for the fluorescence correlation spectroscopy correlation function. The dependence of this function on various model parameters, such as chain length, persistence length, and fluorescence label density, is discussed. Our investigations show that the intramolecular dynamics provides a significant contribution or even dominates the correlation function as soon as the longest intramolecular relaxation time significantly exceeds the shortest experimentally accessible time. Correspondingly, the shape of the correlation function changes considerably. Approximate analytical expressions are provided, which are in qualitative agreement with the exact theoretical solutions as well as experimental results, for both DNA and actin filaments. Our approach is in agreement with the predictions of the Zimm model, in the limit of very flexible polymers, as well as the predictions of semiflexible polymer models with respect to the intramolecular dynamics in solution.

Figure 1

Diffusive parts of the FCS correlation functions for continuously labeled (flexible) polymers of various lengths: $pL={10}^1$, $2\times {10}^1$, $5\times {10}^1$, ${10}^2$, $2\times {10}^2$, $5\times {10}^2$, ${10}^3$, $2\times {10}^3$, $5\times {10}^3$, and ${10}^4$ (left to right). Other parameters are $l_p=50\mathrm{nm}$, $d=2.5\mathrm{nm}$, $T=300\mathrm{K}$, $\eta =0.01\mathrm{P}$, $r_0=0.18\mu \mathrm{m}$, and $f=7$. The inset shows the FCS correlation functions calculated with intramolecular contributions only ($pL$ increases from top to bottom).

Figure 2

FCS correlation functions for (flexible) polymers of various lengths ($pL$ values see Fig. 1). The dashed lines are calculated taking into account intramolecular contributions only ($pL$ increases from right to left).

Figure 3

Slopes of the functions $\log [1∕G_{\mathrm{D}}\left(t\right)-1]$ for various polymer lengths ($pL$ values as in Fig. 1). The solid lines correspond to the total correlation function and the dotted lines to the intramolecular contribution ($pL$ increases from left to right).

Figure 4

Comparison of the full numerically determined FCS correlation function (solid lines) with the analytical approximation of Eq. (45) (dotted lines) for the polymer lengths $pL={10}^2$, $2\times {10}^2$, $5\times {10}^2$, ${10}^3$, $2\times {10}^3$, $5\times {10}^3$, and ${10}^4$. (For the other parameters, see Fig. 1.) The time scale of the analytical solution is multiplied by 1.7 to match the time scale of the full numerical correlation function. Inset: Correlation functions taking into account intramolecular contributions only for $pL={10}^3$ and ${10}^4$ (solid lines: theoretical expression, dotted lines: analytical approximation). The time scale of the analytical solution is multiplied by 1.9 ($pL$ increases from left to right).

Figure 5

Comparison of theoretical FCS correlation functions (solid lines) with experimental measurements (squares) on densely labeled DNA for $pL=9$ $\left(2686\mathrm{bp}\right)$, 25 $\left(7250\mathrm{bp}\right)$, and 165 $\left(48502\mathrm{bp}\right)$ (left to right). The dotted lines represent intramolecular contributions only ($pL$ increases from top to bottom).

Figure 6

Comparison of the theoretical FCS correlation function (solid lines) with experimental measurements on DNA (squares) for $pL=9$, 25, and 165 (left to right). The dotted lines represent intramolecular contributions only ($pL$ increases from bottom to top). Insert: Slopes of the theoretical correlation functions $\log [1∕G_{\mathrm{D}}\left(t\right)-1]$.

Figure 7

FCS correlation functions for various label densities: $N=1,2,3,5,7,11,21$, continuum (left to right). The dotted lines give the intramolecular contributions only. The chain length is $pL=200$. The other parameters are the same as in Fig. 1. Inset: Slopes of $\log (1∕G_{\mathrm{D}}-1)$ for various label densities (increasing from right to left). The slopes for $N=51$ are presented in addition to those given above.

Figure 8

FCS correlation functions for (rigid) polymers of various length: $pL=0.0085,0.017,0.035,0.085,0.17,0.35,0.85,1.7,3.5$ (left to right). Further parameters are: $l_p=16\mu \mathrm{m}$ and $d=8\mathrm{nm}$. The remaining parameters are the same as in Fig. 1. The inset shows the FCS correlation functions in a semilogarithmic representation.

Figure 9

FCS correlation functions for (rigid) polymers of various lengths (increasing from right to left). Solid lines: total correlation functions; dotted lines: intramolecular contributions. The parameters are the same as in Fig. 8.

Figure 10

Slopes of the function $\log [1∕G_{\mathrm{D}}\left(t\right)-1]$ for various polymer lengths (increasing from left to right; $pL$ values as in Fig. 8). The solid lines correspond to the total correlation function and the dotted lines to the intramolecular contributions only.

Figure 11

Comparison of the full theoretical FCS correlation function (solid lines) to the analytical approximation of Eq. (50) (dotted lines) for various polymer lengths. The time scales of the analytical approximations are multiplied by the following factors to match the time scales of the full numerical correlation functions: 1.0 $(pL=0.0085)$, 0.95 (0.017), 1.0 (0.035), 1.2 (0.085), 1.3 (0.17), 1.4 (0.35), 1.5 (0.85), and 1.7 (1.7) (left to right). (For the other parameters see Fig. 8.)

Figure 12

Comparison of the theoretical FCS correlation functions (solid lines) to experimental measurements (squares) on actin filaments for $pL=0.0036$, 0.018, 0.035, and 0.15 (left to right). Inset: FCS correlation function for $pL=0.85$. The dotted line corresponds to the intramolecular contribution only.