Connecting Longitudinal and Transverse Relaxation Rates in Live-Cell NMR

Abstract

In the cytosolic environment, protein crowding and Brownian motions result in numerous transient encounters. Each such encounter event increases the apparent size of the interacting molecules, leading to slower rotational tumbling. The extent of transient protein complexes formed in live cells can conveniently be quantified by an apparent viscosity, based on NMR-detected spin-relaxation measurements, that is, the longitudinal (T₁) and transverse (T₂) relaxation. From combined analysis of three different proteins and surface mutations thereof, we find that T₂ implies significantly higher apparent viscosity than T₁. At first sight, the effect on T₁ and T₂ seems thus nonunifiable, consistent with previous reports on other proteins. We show here that the T₁ and T₂ deviation is actually not a inconsistency but an expected feature of a system with fast exchange between free monomers and transient complexes. In this case, the deviation is basically reconciled by a model with fast exchange between the free-tumbling reporter protein and a transient complex with a uniform 143 kDa partner. The analysis is then taken one step further by accounting for the fact that the cytosolic content is by no means uniform but comprises a wide range of molecular sizes. Integrating over the complete size distribution of the cytosolic interaction ensemble enables us to predict both T₁ and T₂ from a single binding model. The result yields a bound population for each protein variant and provides a quantification of the transient interactions. We finally extend the approach to obtain a correction term for the shape of a database-derived mass distribution of the interactome in the mammalian cytosol, in good accord with the existing data of the cellular composition.

Figure 1

Probe proteins in-cell NMR properties and relaxation. (A) shows the proteins TTHA^pwt (blue), HAH1^pwt (red), and SOD1^barrel (green) including secondary structure elements. The electrostatic surface of each protein is displayed, with blue-colored patches belonging to the basic, positively charged residues arginine or lysine and the red colored patches belonging to the acidic, negatively charged residues glutamate or aspartate. The surface charge mutations are highlighted as black spheres. (B) depicts HMQC spectra of the reporter protein electroporated into live A2780 cells. (C) In-cell NMR relaxation data of the three basis reporter proteins are shown. Signal intensity attenuation obtained from the R₁ (dark gray) and R₂ (light gray) experiments are shown as filled circles and the corresponding fitted single exponential fits are shown as solid lines. The error bars in the relaxation rates are estimated from the signal-to-noise-ratio in each experiment (Supporting Information Methods).

Figure 2

NMR relaxation data and apparent viscosity derived therefrom. (A,B) NMR relaxation rates as functions of rotational correlation time, τ_r, and molecular weight M_w, at 700 MHz (16.5 T) field strength. The colored circles are the measured relaxation rates [(A) R₁, (B) R₂] for the three proteins (HAH1^pwt: red, SOD1^barrel: green and TTHA^pwt: blue) in increasingly viscous glycerol solutions. While τ_r values calculated from the two relaxation rates (Supporting Information 1) fit the predicted theoretical values (black line) in the case of the in vitro glycerol data, clear deviations from the theory are found for the in-cell relaxation data (triangles), where the brighter colors correspond to the respective surface mutation. (C) Apparent viscosities η^app derived from transverse, R₂ (squares), and from longitudinal in-cell relaxation, R₁ (circles), plotted against protein net charge, where the colors are as in (A,B). The lines are empirically fitted exponential curves, with an offset corresponding to the intrinsic viscosity of water. The marked discrepancy between the obtained η^app values suggests that the mean-field η^app model is insufficient for explaining changes in NMR relaxation due to intracellular encounters.

Figure 3

Agreement between observed and calculated reduced relaxation rates for the different models. The calculated R₁ = T₁^–1 values in (A) are from the mean field η^app approach, note the different axis scale in this figure. (B) Corresponds to fast exchange to a binding partner with an optimized mass of 143 kDa. In (C, D), lognormal mass distributions are used, where the shape factors are optimized in (D). The dashed line corresponds to a 1:1 correlation. Reduced R₁ values are shown as circles, while reduced R₂ values are depicted as squares. The color coding is the same as in Figures 1 and 2.

Figure 4

Size distribution estimated from in-cell relaxation data agrees well with the database-derived size distribution of cytosolic proteins. (A) Normalized histogram representation of the mass distribution of the cytosolic proteome of human proteins from the Uniprot database, with 5217 proteins. A lognormal probability density function was fitted to the histogram (orange line). (B) Net charge density shows an approximately normal distribution centred at −0.065 e/nm². The dashed line corresponds to zero charge. (C) Optimized size distributions combining R₁ and R₂ data from the six reporter proteins into a single binding model. The optimized lognormal distribution (black) compared to the database-derived distribution (orange), corresponding to the fitted distribution in (A). The gray shaded area depicts the variation upon random removal of one (dark gray) or two (bright gray) relaxation pairs. The dashed gray line is the maximum entropy distribution from the family of solutions (Supporting Information 6, Figures S8, S9).

Figure 5

Determined bound population as a function of net charge. The optimized population from a model with transient interaction of the reporter proteins with a distribution of interaction partners, blue marker: TTHA^pwt; red marker: HAH1^pwt; and green marker: SOD1^barrel. The brighter markers correspond to the surface mutation variants of the reporter proteins and highlight the importance of surface net charge on in-cell transient encounter formations.

Figure 6

Comparing the models. The mean field approach, where a reporter protein (orange) is assigned an apparent mass (blue panel), cannot simultaneously describe in-cell R₁ and R₂ data. However, a model with the reporter protein in a single free state and a population in a distribution of bound states fully reconciles the R₁ and R₂ data. The distribution of monomeric cytosolic proteins is not sufficient by itself but transient interactions with larger components, such as protein assemblies, membranes, ribosomes, and cytoskeleton have to be accounted for.