^{1,a)}and Dmitrii E. Makarov

^{1,2,b)}

### Abstract

We describe a two-dimensional (2D), four-color fluorescence resonance energy transfer (FRET) scheme, in which the conformational dynamics of a protein is followed by simultaneously observing the FRET signal from two different donor-acceptor pairs. For a general class of models that assume Markovianconformational dynamics, we relate the properties of the emission correlation functions to the rates of elementary kinetic steps in the model. We further use a toy folding model that treats proteins as chains with breakable cross-links to examine the relationship between the cooperativity of folding and FRET data and to establish what additional information about the folding dynamics can be gleaned from 2D, as opposed to one-dimensional FRET experiments. We finally discuss the potential advantages of the four-color FRET over the three-color FRET technique.

We are grateful to Rick Russell, Ben Schuler, and Haw Yang, for helpful discussions. This work was supported by the National Science Foundation (Grant No. CHE 0347862) and the Robert A. Welch Foundation (Grant No. F-1514).

I. INTRODUCTION

II. THEORY

A. A simple example: Two weakly correlated FRET pairs

B. 2D FRET statistics for the general case

III. FRET DYNAMICS IN THE GCCL MODEL

A. Description of the model

B. Case study: Structure 1

C. Case study: Structure 2

IV. CONCLUDING REMARKS

### Key Topics

- Protein folding
- 23.0
- Proteins
- 22.0
- Correlation functions
- 15.0
- Polymers
- 15.0
- Conformational dynamics
- 13.0

## Figures

Folding of a two-domain protein as manifested by 2D FRET, where each domain has been tagged with a donor-acceptor FRET pair. (A) For uncoupled domains, states of the protein form a rectangle in the plane. (B) Kinetic coupling between conformational dynamics of each domain alters the rate constants of folding and unfolding; structural coupling between two domains alters their FRET efficiencies such that the states no longer form a rectangle.

Folding of a two-domain protein as manifested by 2D FRET, where each domain has been tagged with a donor-acceptor FRET pair. (A) For uncoupled domains, states of the protein form a rectangle in the plane. (B) Kinetic coupling between conformational dynamics of each domain alters the rate constants of folding and unfolding; structural coupling between two domains alters their FRET efficiencies such that the states no longer form a rectangle.

Correlation functions for two coupled domains. The cross-correlation function is plotted for , (thin solid line), , (dashed line), and , (dotted line). See text for the definition of the coupling parameters. The heavy solid line shows the autocorrelation functions for all three cases, which are indistinguishable in this plot.

Correlation functions for two coupled domains. The cross-correlation function is plotted for , (thin solid line), , (dashed line), and , (dotted line). See text for the definition of the coupling parameters. The heavy solid line shows the autocorrelation functions for all three cases, which are indistinguishable in this plot.

A cartoon depiction of structure 1 (top) and structure 2 (bottom). Native contact points are shown as open circles. FRET donor and acceptor dyes are shown as filled circles.

A cartoon depiction of structure 1 (top) and structure 2 (bottom). Native contact points are shown as open circles. FRET donor and acceptor dyes are shown as filled circles.

Top: The average FRET efficiency as a function of for the donor-acceptor pair {1,61} of structure 1. Bottom: The characteristic relaxation times (points), (dashed line), and (solid line) as a function of for structure 1. See Eqs. (32)–(34) for the definition of these times.

Top: The average FRET efficiency as a function of for the donor-acceptor pair {1,61} of structure 1. Bottom: The characteristic relaxation times (points), (dashed line), and (solid line) as a function of for structure 1. See Eqs. (32)–(34) for the definition of these times.

Contour plot of the joint probability distribution of the FRET efficiency at for structure 1. The FRET pairs are defined by and . Also shown are the points for each state and straight lines showing the connectivity of all possible paths. The lines and provide provide examples of elementary transitions with and without structural coupling between the FRET pairs, respectively (see Fig. 6).

Contour plot of the joint probability distribution of the FRET efficiency at for structure 1. The FRET pairs are defined by and . Also shown are the points for each state and straight lines showing the connectivity of all possible paths. The lines and provide provide examples of elementary transitions with and without structural coupling between the FRET pairs, respectively (see Fig. 6).

A cartoon showing the rearrangements of the chain in the transition and highlighted in Fig. 5, with dye molecules shown as filled circles.

A cartoon showing the rearrangements of the chain in the transition and highlighted in Fig. 5, with dye molecules shown as filled circles.

Top: The correlation functions (solid line), (dotted line), and (dashed line) plotted for structure 1 at . The FRET pairs are defined by and . Bottom: Same correlation functions plotted on a semilogarithmic scale. The heavy solid line is plotted vs , where is the lowest nonzero eigenvalue of the matrix .

Top: The correlation functions (solid line), (dotted line), and (dashed line) plotted for structure 1 at . The FRET pairs are defined by and . Bottom: Same correlation functions plotted on a semilogarithmic scale. The heavy solid line is plotted vs , where is the lowest nonzero eigenvalue of the matrix .

Top: The correlation functions (solid line), (dotted line), and (dashed line) plotted for structure 1 at . The FRET pairs are defined by and . Bottom: Same correlation functions plotted on a semilogarithmic scale. The heavy solid line is plotted vs , where is the lowest nonzero eigenvalue of the matrix . Note the the cross-correlation function is indistinguishable from .

Top: The correlation functions (solid line), (dotted line), and (dashed line) plotted for structure 1 at . The FRET pairs are defined by and . Bottom: Same correlation functions plotted on a semilogarithmic scale. The heavy solid line is plotted vs , where is the lowest nonzero eigenvalue of the matrix . Note the the cross-correlation function is indistinguishable from .

Contour plot of the joint probability distribution of the FRET efficiency at for structure 2. The FRET pairs are defined by and . Also shown are the points for each state and straight lines showing the connectivity of all possible paths.

Contour plot of the joint probability distribution of the FRET efficiency at for structure 2. The FRET pairs are defined by and . Also shown are the points for each state and straight lines showing the connectivity of all possible paths.

Top: The correlation functions (solid line), (dotted line), and (dashed line) plotted for structure 2 at . The FRET pairs defined by and . Bottom: Same correlation functions plotted on a semilogarithmic scale. The heavy solid line is plotted vs , where is the lowest nonzero eigenvalue of the matrix .

Top: The correlation functions (solid line), (dotted line), and (dashed line) plotted for structure 2 at . The FRET pairs defined by and . Bottom: Same correlation functions plotted on a semilogarithmic scale. The heavy solid line is plotted vs , where is the lowest nonzero eigenvalue of the matrix .

## Tables

The correlation coefficients for structure 1, where and are for FRET pairs attached to the monomers of the corresponding row and column contacts of the table element.

The correlation coefficients for structure 1, where and are for FRET pairs attached to the monomers of the corresponding row and column contacts of the table element.

for structure 1, where and are for FRET pairs attached to the monomers of the corresponding row and column contacts of the table element.

for structure 1, where and are for FRET pairs attached to the monomers of the corresponding row and column contacts of the table element.

The correlation coefficients for structure 2, where and are for FRET pairs attached to the monomers of the corresponding row and column contacts of the table element.

The correlation coefficients for structure 2, where and are for FRET pairs attached to the monomers of the corresponding row and column contacts of the table element.

for structure 2 where and are for FRET pairs attached to the monomers of the corresponding row and column contacts of the table element.

for structure 2 where and are for FRET pairs attached to the monomers of the corresponding row and column contacts of the table element.

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