IV. Ultrafast Biochemistry

Initial events and fluctuations in biochemical processes at the atomic scale

  • Time and length scales of biochemical reactions
  • Photo-initiation of biochemical processes
  • Time-resolved measurement techniques
  • The photocycle of bacteriorhodopsin
  • Dynamics of protein folding and catalytic action
  • Mesoscopic non-equilibrium thermodynamics
Sub-nanosecond processes are fundamental to biochemistry and will be accessible to the ultrashor t pulses of the SwissFEL. Impor tant categories of ultrafast natural and ar tificial photoexcited bio-reactions exist, and the phenomenon of molecular caging offers the possibility of rapidly photo-triggering wide classes of biochemical phenomena. Although a typical bio-reaction, involving the coordinated motion of large protein subunits, may require a millisecond to complete, the reaction occurs on a complex energy landscape, passing through a continuum of biologically-relevant intermediate states. The temporal and spatial resolution provided by the SwissFEL will allow the exploration of molecular biochemical trajectories on these energy landscapes, both as irreversible pump-probe events or as equilibrium fluctuations. Fresh insights into the biochemical processes at shor t time and length scales, compatible with the SwissFEL can be won by applying the developing formalism of meso-scale non-equilibrium thermodynamics.

Time and length scales of biochemical reactions

The biological length scales per tinent to SwissFEL applications span the spectrum from individual atomic bonds and small molecules, such as glucose, in the subnanometer range, over individual protein molecules (e.g., hemoglobin) and molecular complexes (e.g., ribosomes) in the 1–10 nm range, over virus par ticles and cell organelles (10–100 nm), to entire cells (1 μm) (see Fig. IV.1). As discussed in Chapter III, with hard radiation from the SwissFEL (λ = 0.1 nm), lensless imaging of nanostructures is feasible at sub-nm resolution. As the sampled objects grow larger and more complex, the realistically achievable resolution degrades, to perhaps several tens of nm for micrometer-sized cells. The vibration period of the two carbon atoms in the ethene molecule against their mutual double bond is 20 fs, and such molecular vibrations also set the time scale for photo-dissociation processes, for example for the photo-detachment of CO from a heme group (i.e., an iron atom surrounded by a porphyrin ring). As biomolecular units increase in size, from methyl groups (0.3 nm) to side-chains (1 nm) to loops (4 nm), to large protein domains (8 nm), collective motions with long periods populate the crowded vibrational density of states. Although electron transfer processes can in principle be very fast, their speed is generally limited by that of molecular reconformation. The same is true for decaging processes – the activation of a biomolecule by the photo-removal of a deactivating cage group.

Photo-initiation of biochemical processes

The present Chapter concerns itself with the investigation of ultrafast biochemical processes with the Swiss- FEL. Since most such studies will be per formed in the optical pump – X-ray probe scheme, one needs to ask: which photo-initiated effects can be used as optical pump triggers? Well-known photo-excited biochemical processes in proteins can be classified as “natural” or “ar tificial” [1]. Natural photo-activated protein-components are: the photosynthetic reaction center of chlorophyll (causing light-harvesting and electron transfer), the retinal group in rhodopsin vision complexes (causing isomerization, proton pumping and membrane polarization), the flavin group in DNA photolyase (per forming DNA repair in plants by photocatalytically removing pyrimidine dimers) the cryptochrome and phototropin photoreceptors (causing electron transfer and covalent reactions), and the linear tetrapyrroles in phytochrome photoreceptors of plants and bacteria (causing photoisomerization). Artificial photo-activated components are: the flavin group in flavodoxin (causing electron transfer), and the heme group in hemoproteins (e.g., myoglobin) (causing redox chemistry and a so-called “protein quake” – see Fig. IV.2). A major theme in fast and ultrafast biochemistry, discussed below, is the folding of a protein to its native state. Several methods of photo-initiating the folding process have been developed, including: a) rapid temperature jumps, down to 50 ns, which cause, e.g., helix solvation [4], b) ns laser photolysis of folding-inhibiting ligands [5], c) photo-cleavable protein cross-linking reagents [6], and d) folding induced by electron transfer [7]. A fur ther possible fast per turbation of biomolecules in solution is via a ps teraher tz pulse. It is known that THz absorption disrupts the H2O network by breaking hydrogen bonds and that the function of a protein molecule is influenced by its surrounding solvation shell of up to 1000 water molecules [8]. Finally, it is possible to reversibly deactivate a biomolecule by the addition of a caging group [9, 10]. When this group is photocleaved away, by a visible or UV light pulse, the biomolecule becomes activated, via a rapid series of decaging reactions (see Fig. IV.3). A par ticularly attractive possibility is to rapidly decage the biological energy-carrier ATP [11], a feat which has been accomplished, using a Coumarin derivative, yielding a decaging time of 600 fs [12].

Time-resolved measurement techniques

Several methods of studying biochemical dynamics have been developed. Among the low-resolution methods are: optical absorption spectroscopy in the visible and IR regions, circular dichroism and Raman scattering. The hydrogen-deuterium exchange method, based on the different exchange rates of hydrogen isotopes for exposed and hidden amino acids in a protein immersed in D2O, detects folding or unfolding on the time scale of milliseconds or longer. The Fluoresence Resonant Energy Transfer (FRET) method allows a determination of the state of folding of a single protein molecule with a time resolution of the order of 10 ns [13, 14] (see Infobox). Atomic scale information can be obtained by Nuclear Magnetic Resonance on the time scale of 10– 0.1 seconds, by the observation of peak splitting, and of ms – μs, by line broadening. Sychrotron-based, time-resolved Small and Wide Angle X-ray Scattering (TR-SAXS/WAXS) from photo-triggered biomolecules in solution is capable of providing nm spatial resolution and 100 ps time-resolution, limited by the X-ray pulse length (see Fig. IV.4) [15]. Biomolecules are carried by a liquid jet – at the SwissFEL this must be in vacuum – using the technology presented in Chapter II. An optical laser pulse excites the molecules, which are then probed, after a preset time delay, by the 100 ps X-ray pulse. The same experimental station as used for time-resolved SAXS is also employed to per form pump-probe Laue crystallography. For the CO-photo-detachment in crystalline myoglobin, shown in Figure IV.5, broadband X-ray pulses (3% bandwidth, centered at 15 keV photon energy) were used. Each X-ray pulse had 1010 photons, 32 pulses were acquired for each crystal orientation, and 31 different orientations were measured, without laser pump and with various pump-probe delays. With stable, ultra-shor t pulses from the SwissFEL, such time-resolved pump-probe SAXS and Laue experiments will be possible with a much improved 20 fs time resolution. Optimally configured broad-band radiation for timeresolved Laue diffraction will be provided by detuning the individual undulator sections of the SwissFEL, at some cost in intensity.


























Fluorescence Resonant Energy Transfer (FRET)

FRET is an optical spectroscopic technique with which the dynamics of a single biomolecule can be followed (Fig. IV.i1). Incident optical light is absorbed by chromophore 1 at position 1 in the biomolecule. If sufficiently close by, i.e., if the protein is folded, the excitation energy can be transferred to a different chromophore 2, at position 2 in the molecule, resulting in the characteri-stic fluorescence signal of chromophore 2. If the inter-chromophore distance is too great, fluorescence will occur characteristic of chromophore 1. For single molecule FRET, the achievable time resolution is of the order of 10 ns [14]. Although a power ful technique for following the dynamic behavior of a single biomolecule, FRET cannot provide the detailed structural information available with the SwissFEL.

The photocycle of bacteriorhodopsin

As an example of a classic photo-induced biochemical process, we discuss the photocycle of bacteriorhodopsin, a model system for the biological basis of vision. The information presented was obtained using pump-probe optical spectroscopy and X-ray diffraction from flashfrozen samples. Pump-probe SAXS or Laue experiments at the SwissFEL offer the exciting possibility of directly determining the detailed molecular structure of even the shor test-lived intermediate states. In higher organisms, including invertebrates and humans, it is the membrane protein rhodopsin, found in the rod cells of the retina (see Fig. IV.6), which delivers the photon energy to a complex series of biochemical processes, leading ultimately to an electrical polarization of the membrane and hence to a nerve impulse. In 1967, Wald, Granit and Har tline received the Nobel prize for medicine for the discovery of the initial event in vision: a photon-induced trans → cis structural transformation in retinal, the photoreceptor located at the center of the rhodopsin molecule. This transformation is the fastest biological photoreaction known (see Fig. IV.7). Due to a highly complex photocycle and problems of photo-instability, rhodopsin is difficult to handle in the laboratory. Bacteriorhodopsin (bR), extracted from the “purple membrane” of Halobacterium salinarum, is easier to study and has a simpler photocycle (see Fig. IV.8) [17]. However, in spite of the fact that no membrane protein has been studied as extensively as bR, details of its photocycle are still controversial. Recent fs optical spectroscopy experiments have identified two additional shor t-lived intermediate states, not shown in Figure IV.8, which form just after the photoexcitation (see Fig. IV.7, above): The “I-state”, which forms within 200 fs, is believed to incorporate an incomplete (90°) bond rotation, and the “J-state”, with a rise time of 500 fs, may be a vibrationally-excited version of the K state. Finally, high-resolution flash-frozen X-ray diffraction investigations of bR indicate distor tions in the helices of the entire bR molecule [18] (see Fig. IV.9). For bacteriorhodopsin and other photo-sensitive biomolecules, a wealth of information is awaiting the greatly improved spatial and temporal resolutions of the Swiss- FEL.
























Dynamics of protein folding and catalytic action

Synchrotron-based X-ray protein crystallography is the ideal technique for determining the static atomic-scale struture of protein molecules and their complexes, provided sufficient material can be purified and crystallized. But it is the dynamic structure of a protein molecule which determines its function. One can see from Figure IV.1 that major reconformations of proteins are slow processes on the scale of the 20 fs XFEL pulse duration, requiring μs to ms to complete. What are the impor tant questions regarding the folding process [19, 20] to which the SwissFEL can contribute? Since the work of Anfinsen et al, in the early 1960’s [21], it is known that the folding of a protein is a reversible process; the native structure is a thermodynamically stable configuration, corresponding to a global minimum of the accessible free energy. But according to the “Levinthal paradox” (see Infobox), it is vir tually impossible for a protein molecule to sample all of the available structures. Dill and Chan [19] ask the question: “Among a multitude of possibilities, how does a protein find its equilibrium structure?” In its search for the minimum energy state, the protein can be thought to move on an energy sur face in (higherdimensional) conformational space, the shape of which is determined by the amino-acid sequence of the protein and by external factors such as pH, temperature, degree of solvation, and the presence of neighboring “chaperone” proteins (see Infobox). What is the nature of this energy sur face? Several schematic possibilities are shown in Figure IV.10 [19]. The Levinthal paradox is portrayed in Figure IV.10a – the “Levinthal golf course”: the unrealistic absence of an energy gradient presents the protein with an insurmountable entropic barrier to finding the minimum-energy state. The alternative of a narrow “folding pathway“ (Fig. IV.10b) is also deemed unrealsitic, since it fails to describe proteins which happen to land off the beaten road. Current thinking prefers the picture that the protein performs a diffusional motion on the “trickle funnel” landscape shown in Figure IV.10c: all the protein molecules are guided to the native state, albeit with possible detours around local energy maxima and entropic delays in regions of low gradient. It has been proposed that par t of the folding trajectory consists of “hydrophobic zipping” [22, 23]. This is “an opportunistic process in which local contacts (nearby in the sequence) form first, drawing in new contacts, wich create still other and increasingly nonlocal contacts and opor tunities of other intrachain interactions. Helices, turns and other local structures would be the first to zip” [19]. Dill and Chan [19] stress that it would be par ticularly valuable to have an experimental method capable of quantifying the time-dependent structural correlations occuring in an ensemble of identical fluctuating proteins (see Fig. IV.11). This, they argue, would provide important information on the nature of the energy landscape and on the diffusional trajectories followed by the folding and unfolding proteins. The time-space correlation method proposed by Kam (see Chapter III) for studying the static structure of biomolecules in solution could conceivably be extended, using the SwissFEL, to the investigation of such dynamic structural correlations. Examining the protein-folding energy landscape in detail, Henzler-Wildman and Kern [20] classify the energy barriers according to their heights into “tiers” 0, 1 and 2 (see Fig. IV.12). Tier 0 refers to barriers of several kBT, corresponding to “larger-amplitude collective motions between relatively small numbers of states.” Transitions between these states occur on the μs time-scale or slower and form the basis of many impor tant biological processes, including “enzyme catalysis, signal transduction and protein-protein interactions”. On the finer scales of tiers 1 and 2, a “large ensemble of structurally similar states that are separated by energy barriers of less than one kBT result in more local, small-amplitude ps to ns fluctuations at physiological temperature”. It has been determined that these fluctuations, involving, e.g., loop motions (ns) and side-chain rotations (ps), in effect “pre-sample” the characteristic motions of catalytic action. Still finer than tier 2 are the fs-scale fluctuations related to bond vibrations and interactions with the solvent. Henzler-Wildman and Kern differentiate between the large tier 0 “enthalpic barriers”, relevant to conformational conformational change and the “entropic barriers” on the scale of tiers 1 and 2, which are related to minor rearrangements among neighboring peptides and in their interactions with the solvent. Thus, whereas the coherence and high peak flux of the SwissFEL make it a power ful tool for quantifying large conformational changes of proteins, its real power lies in the ability to follow the pre-sampling of biologically impor tant pathways occuring via delicate fs-ps-ns fluctuations. Fundamental questions regarding protein dynamics, which the SwissFEL may help resolve, include [19, 20, 24]:
  • What are the predominant structures and the distance distributions in the unfolded state?
  • What is the “speed limit” for elementar y dynamics along the energy sur face?
  • How rugged is the energy landscape? A rugged landscape implies fluctuations among very different conformations.
  • Is it possible to measure not only averages of structural observables, but also correlations among their fluctuations (Fig. IV.11)?
  • Can minor conformational substates be predicted from known structures?
  • How can this knowledge and a dynamic view of proteins be used to help discover and develop novel therapeutic agents?

Levinthal’s paradox

In the late 1960’s, Cyrus Levinthal formulated the “Levinthal paradox” [27], which can be formulated as a stastical problem [28]: Consider a protein molecule composed of 100 residues, each of which can assume 3 different conformations. For the number of possible structures, this yields 3100 = 5 × 1047. Now assume that it takes (only) 100 fs to convert from one structure to another. It would thus require 5 × 1034 s = 1.6 × 1027 years to (systematically) explore all possibilities. The disagreement between this long time and the actual folding time (μs – ms) represents Levinthal’s paradox.





HANS FRAUENFELDER

born 1922 in Schaffhausen, Switzerland,
PhD from ETHZ
Quote from Ref. [29]:
To carry out their functions, most proteins must perform motions. These motions can either be thermal equilibrium fluctuations or non-equilibrium relaxations, caused for instance by reactions. In terms of the energy landscape, motions can be described as jumps of the system from substates to substates. A task of biological physics is the experimental study of these motions, their connections to structure, and to the energy landscape. Since the rate coefficients of fluctuations range from fs-1 to s-1 or possibly even less, it is clear that many different tools are needed. It has been known for some time that proteins share proper ties with glasses. Recent studies show that this similarity has unexpected aspects, related to the interaction of the protein proper with the bulk solvent and with its hydration shell. Consider first the large-scale fluctuations. A comparison of the rate coefficients, kp(T), for large-scale motions in proteins, for instance entry and exit of ligands, with the rate coefficient kα(T) of the bulk solvent shows that they have the same temperature dependence over many orders of magnitude. In other words, large-scale fluctuations of the protein are slaved to the fluctuations in the bulk solvent. These protein processes are controlled by enthalpy barriers in the solvent, not by protein-internal enthalpy barriers. But a puzzle appears: while kp(T) and kα(T) have the same temperature dependence, for some processes kp(T) is 105 times slower than kα(T)! What causes this slowing? It obviously must be entropy or, in other words, the large number of states in the protein. Here is where the energy landscape comes in. A process like the exit of a ligand is not like opening a rigid door. Many side chains must be in the right position, helices may have to move. Opening thus corresponds to a random walk in conformation space. Indeed, theory suppor ts such a picture. Significant proper ties of proteins follow from these experimental results: proteins work in close interaction with their environment, the environment controls the enthalpy barriers for large-scale motions; the protein proper contributes to the entropy as characterized through the energy landscape. A complete understanding of the energy landscape and of the related fluctuation and relaxation processes is still a dream, but it is a grand challenge for biological physics.

Mesoscopic non-equilibrium thermodynamics

It is clear from the discussion of the previous Section that issues of fluctuations and entropy, on shor t time and length scales, play an impor tant role in ultrafast biochemistry. Fur thermore, a biological system is of necessity in a state of non-equilibrium; “equilibrium” is a synonym for “death”. The question arises, is there a general formalism, on a level of sophistication above that of statistical mechanics, for treating a non-equilibrium state on the scale of macro-molecular fluctuations? Such a formalism, Mesoscale Non-Equilibrium Thermodynamics (MNET), is being developed for just this purpose [25]. Classical thermodynamics provides general rules for continuum, macroscopic systems in thermal equilibrium, in terms of a few parameters, such as the extensive variables internal energy, volume and mass, and intensive variables such as temperature, pressure and chemical potential. The dependence of the entropy on these variables is given by the Gibbs equation:
TdS = dE + pdV – μdM
(equilibrium thermodynamics) In a non-equilibrium situation, gradients in the intensive variables act as forces, causing extensive fluxes. Such cases are treated in the framework of non-equilibrium thermodynamics (NET), in which the system is considered to consist of many subsystems, each of which is described, still in a continuum approximation, by equilibrium thermodynamic variables and each of which experiences a non-negative change in entropy. Consider the simple case of one-dimensional mass diffusion at constant energy and volume (dE = 0, dV = 0). In NET, the Gibbs equation for non-equilibrium thermodynamics may now incorporate a chemical potential with explicit and implicit spatial dependences, and hence the expression for the entropy production should read (see Infobox):
T\frac{\partial S}{\partial t} = - \int \mu [x, \rho(x)]
\frac{\partial \rho(x)}{\partial t} dx ~~ (NET)
Under cer tain assumptions (see Infobox), one obtains the well-known diffusion equation for the density:
\frac{\partial \rho}{\partial t} = \frac{\partial}{\partial
x} D \frac{\partial \rho}{\partial x}
where D = L \frac{\partial \mu}{\partial \rho} is the diffusion constant, and L is the Onsager coefficient. Although NET allows the description of irreversible processes, the fact that it still is based on a continuum picture makes it inappropriate to treat systems at the molecular level, where strong fluctuations are important. Fur thermore, the defining relationships, such as the dependence of the fluxes on the forces, are linear. This makes it impossible for NET to describe, for example, activated processes in biochemistry. To extend NET to very shor t time and length scales (accessible with the SwissFEL), the statistical approach of Mesoscopic Non- Equilibrium Thermodynamics (MNET) is called for. The star ting point is again the dynamics of the entropy of a system, this time based on the Gibbs entropy postulate:
S = S_{eq} - k_B \int P(\gamma, t) \ln \frac{P(\gamma,
t)}{P_{eq}(\gamma)} d\gamma ~~ (MNET)
which relates the deviation of entropy from its equilibrium value with the probability density P of finding the system in a state with par ticular values for its nonequilibrated degrees of freedom γ. These degrees of freedom could include, e.g., a par ticle velocity, a spin orientation or the conformation of a biomolecule. The chemical potential is now µ(γ,t) =  (γ,t) + kT lnP(γ,t), where  is the activation energy barrier in an internal space coordinate at equilibrium, and the flux is again a linear function of the driving force
J(\gamma) = -L(\frac{kT}{P} \frac{\partial P}{\partial \gamma} +
\frac{\partial \Phi}{\partial \gamma}) ~~ (MNET)
The NET expression for entropy production then leads to the Fokker-Planck equation
\frac{\partial P(\gamma)}{\partial t} = \frac{\partial}{\partial
\gamma}(D \frac{\partial P}{\partial \gamma} + \frac{D}{kT}
\frac{\partial \Phi}{\partial \gamma}) ~~ (MNET)
where the diffusion constant is now D = \frac{k_B L(\gamma,P)}{P} and L(γ,P) is again an Onsager coefficient, this time dependent on the thermodynamic state P and the mesoscopic coordinates γ. Here one has an explicit formalism to treat on equal footing an enthalpic and an entropic barrier. In a manner similar to the dissipation-fluctuation theorem (see Chapter I), MNET provides a method to “determine the dynamics of a system from its equilibrium proper ties”. Mesoscopic Non-Equilibrium Thermodynamics has application in the study of transport processes in confined systems, such as those in the biochemistry of the cell [26]. “Transport at the mesoscale is usually affected by the presence of forces of different nature: direct interactions between par ticles, hydrodynamic interactions mediated by the solvent and excluded volume effects. The presence of such diversity of forces has a direct implication in the form of the energy landscape, which may exhibit a great multiplicity of local minima separated by barriers. Transpor t at those scales presents two main characteristics: it is intrinsically non-linear, and it is influenced by the presence of fluctuations, external driving forces, and gradients. MNET can be used to infer the general kinetic equations of a system in the presence of potential barriers, which in turn can be used to obtain the expressions for the current of par ticles and the diffusion coefficient” [25]. The fields of application of MNET, namely molecular systems undergoing rapid fluctuations, match well those of the SwissFEL, and it is anticipated that the two could form a tight symbiotic relationship of theory and experiment.

Non-Equilibrium Thermodynamics (NET)

To derive the diffusion equation in the NET formalism [25] (where, for example, the chemical potential μ and the particle density ρ may depend on the position x), one begins with the Gibbs equation for entropy change, assuming constant temperature, energy and volume:
T \frac{\partial S}{\partial t} = - \int \mu[x, \rho(x)]
\frac{\partial \rho(x)}{\partial t} dx
Introducing the mass flux J via the conservation law \frac{\partial \rho}{\partial t} = \frac{\partial J}{\partial x} and per forming integration by par ts (with J vanishing at the system boundary), one obtains:
T \frac{dS}{dt} = - \int J \frac{\partial \mu}{\partial x} dx
This expression establishes that J and \frac{\partial \mu}{\partial x} are “conjugate” quantities, which, according to Onsager, are related by a linear flux-force relation:
J = - L \frac{\partial \mu}{\partial x}
where L is the Onsager coefficient. Inser ting this expression into the conservation law yields the diffusion equation for the density:
\frac{\partial \rho}{\partial t} = \frac{\partial}{\partial x} D \frac{\partial \rho}{\partial x}
where D = L \frac{\partial \mu}{\partial \rho} is the diffusion constant.

Protein folding and human health

The protein-folding problem has a direct relationship to human health (see Fig. IV.i2). “Misfolded and aggregated proteins may be involved in pathological conditions such as Alzheimer’s disease, cystic fibrosis and cataracts. The infectious agent (or prion) in the neurodegenerateive diseases scrapie (in sheep), bovine spongiform encephalopathy (BSE, or “mad cow” disease), and Creutzfeld-Jakob disease (in humans) is also thought to be a misfolded protein” [30].












Summary

  • Important biochemical reactions that require μs to ms to run to completion are composed of more elementary steps on the sub-ns scale.
  • Ultrafast events, on the time scale of fs to ps, occur in photo-initiated reactions, such as photosynthesis, vision and DNA repair, and as fluctuations during slower reactions, which effectively “pre-sample” the relevant energy landscape.
  • The structurally-sensitive X-ray techniques of small-angle scattering and Laue diffraction, when performed in a pump probe manner, provide atomic-level detail not available with optical spectroscopy.
  • The textbook example of an ultrafast phototriggered bioreaction, the photocycle of bacteriorhodopsin, contains many unanswered questions regarding ultrafast structural changes, and it points to similar questions in other impor tant photoreactions.
  • Proteins manage to efficiently find their preferred native state, in spite of an astronomic number of possible configurations. Explaining in detail how this occurs requires experimental methods which are sensitive to local dynamic structural correlations.
  • A rigorous theoretical foundation which accounts for the statistical nature of mesoscale chemical systems on short time scales, “mesoscopic non-equilibrium thermodynamics”, is currently being developed and offers a promising framework for interpreting future SwissFEL experiments in biochemistry