8
Use of Inverse Theory Algorithms in the Analysis
of Biomembrane NMR Data
Edward Sternin
Summary
Treating the analysis of experimental spectroscopic data as an inverse problem and using regularization techniques to obtain stable pseudoinverse solutions, allows access to previously unavailable level of spectroscopic
detail. The data is mapped into an appropriate physically relevant parameter space, leading to better qualitative
and quantitative understanding of the underlying physics, and in turn, to better and more detailed models. A brief
survey of relevant inverse methods is illustrated by several successful applications to the analysis of nuclear magnetic resonance data, yielding new insight into the structure and dynamics of biomembrane lipids.
Key Words: Alignment; de-Pakeing; hexagonal; lamellar; model membranes; order parameter; phase transitions; phospholipid; relaxation rate; Tikhonov regularization.
1. Introduction to the Mathematics of Ill-Posed Problems
Some of the consequences of widely available and ever-increasing computer power are obvious: faster and more detailed data acquisition, more elaborate display capabilities, and an ability
to search through vast arrays of data. However, the ubiquity of computer power can also influence the conceptual aspects of data analysis, changing the way new problems are approached.
Enhanced numerical ability to transform the observed experimental data into forms that are better suited for conceptual analysis yields a profound change in the way the data is modeled and
analyzed. In turn, this better footing for the conceptual thinking leads to better models, and
through making it obvious what experimental aspects need improvement, to better data.
In a traditional research paradigm, experimental data is kept scrupulously separate from the
theoretical model, to avoid biasing the observer. However, converting measured data into
another, more convenient form is acceptable; the Fourier transform (FT) being an excellent
example. Inverse theory methods extend this idea to the possibility of the nature of this transformation itself being a “parameter of the fit,” in some sense. In many situations, the regularizing
influence of real physical constraints is formally added to the inverse algorithm, and ensures
that a physically meaningful transformation is obtained in the end. The process of finding and
testing the optimal transform algorithm can be long and difficult, and it may not always work.
However, it has become clear that for a wide range of problems that one encounters in the
interpretation of spectroscopic nuclear magnetic resonance (NMR) data, this works reliably
and without bias.
The resulting enrichment of the kind of questions one can ask about the data has had a
great impact on several significant problems of current interest in the study of biological and
From: Methods in Molecular Biology, vol. 400: Methods in Membrane Lipids
Edited by: A. M. Dopico © Humana Press Inc., Totowa, NJ
103
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model membrane systems. In this section, a brief introduction to the mathematics of the
inverse problems is presented. Many excellent books on inverse problems exist; the goal here
is not to reproduce their material but to introduce the language of the inverse problems and
to illustrate the effectiveness of inverse methods. To this end, a conceptually simple example
of multiexponential analysis of simulated relaxation data (signals decaying in time) is used
initially. In subsequent sections, practical biomembrane examples are considered, and a successful extraction of a parameter distribution function is used as a stepping-stone toward a
better understanding of the physical properties of each system.
1.1. Indirectly Observed Data
Solving an integral equation∗
f ( y) =
∫ g( x ) K ( y, x )dx
(1)
where f (y) is the experimentally measured quantity (e.g., for an NMR spectrum, y would
have the physical meaning of frequency), g(x) is the desired set of parameters to be determined (e.g., chemical shifts), and K(y, x) is a kernel function that describes the way the
experimental data depends on these parameters is, in general, a mathematically ill-posed
problem. Only a few select kernel functions allow for a complete unambiguous inverse
calculation,
f ( y) → g ( x )
Again, FT is an example. For others, including most of the kernels relevant to NMR, there is
no analytical solution to the inverse problem, and numerical approximate solutions are the
only practical alternatives.
Ill-posedness in fact has a formal definition (1); briefly, the three essential conditions of
ill-posedness are existence, uniqueness, and stability. However, strategies have been developed to combat all three.
Existence: a solution g(x) may not exist at all; the way out is to be explicit about what is
the question that one is asking, and to ask not for the “true” g(x), but for “a reasonable
approximation g̃(x).” Here, “reasonable” is in the least-squares sense, a minimum of misfit
norm
Ψ( g% ) = f ( y) −
∫ g% ( x )K ( y, x )dx
2
→ min
(2)
This misfit norm, the “distance” between the measured f (y) and the approximation calculated
as an integral over g% ( x ) , is sometimes called the least-squares error norm. A minimum
of misfit ensures compatibility of the fit with the measured data.
Uniqueness: many g% ( x ) may satisfy Eq. 1 equally well; the best strategy is to bring in
additional physical input. Of all compatible solutions, choose the one that best satisfies
this additional constraint. Depending on the problem the optimal constraint criteria may
take different forms, but several important criteria are quite universal and apply on very basic
physical grounds. For example, a reasonable physical function should be “smooth” (i.e., locally
*When the limits of integration are fixed as appropriate for the majority of problems of practical interest, this
is the so-called Fredholm integral equation (FIE) of the first kind.
Inverse Algorithms in NMR of Biomembranes
105
well-approximated by a linear function), and minimizing the norm of the second
derivative g″( x ) in addition to the misfit Ψ, provides an universal and very useful criterion for
the selection of the best of all possible solutions.
Stability: a small perturbation in f (y) (e.g., experimental noise) may cause a large
change in g% ( x ). This aspect of ill-posedness turns out to be the most difficult one to deal
with. The solutionrequires regularization of the problem, which enforces local stability of
the f ( y) → g( x ) mapping.
Regularization is the key to finding the inverse solution and it deserves a special detailed
discussion, but first it should be noted that so far only the continuous spaces of both the
experimental variable y and of the parameter-space variable x have been dealt with. In practice, a set of experimental measurements is typically taken as a discrete set of data points, and
so it is useful at this point to formulate a discrete version of the inverse problem. In the language of matrices,
where vectors f = {fi} = {f(yi), i = l … m} and g = {gj} = {g(xj), j = l … n}, are related through
the matrix K = {Kij} = {K(yi, xj), i = 1 … m, j = l … n}. Note how the dimensions of the problem need not be equal (an even-determined problem); depending on the number of available data
points fi and the number of “parameters of the fit” gj, the inverse problem may end up being
over- or underdetermined. For an experimentalist, it is somewhat startling to think of a fit to
more parameters than there are measured data points, but this simply underscores the fact that
regularized solutions of inverse problems are fundamentally different from least-squares fits.
In the matrix context, the inverse problem is equivalent to calculating the inverse matrix,
K−1 such that g = K−1f. If such an inverse does not exist (a majority of cases), the problem
becomes the calculation of the pseudoinverse, the subject of a large and well-developed field
of mathematics. Unfortunately, its significant mathematical complexity represents a formidable challenge to a nonspecialist, and many day-to-day experimental difficulties may dramatically influence the choice of numerical strategies. A small part of the field of inverse
problems will be explored with a very specific goal of implementing some of these techniques as suitable for the problems of biomembrane NMR. As it turns out, selecting the right
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kind of numerical strategy is as important as the specifics of a linear algebra algorithm, and
much depends on the experience and judgment of the person performing the calculation. In
short, “successful inverse problem solving is strongly dependent on the analyst” (2).
1.2. Regularization
The key to finding a pseudoinverse solution is regularization, and to examine various ways
in which a pseudoinverse problem can be regularized, the following illustrative example will
be introduced:
f (t ) =
∫
r max
rmin
g(r ) e− r t dr
(3)
where the generalized x, y variables have been replaced with r, t as appropriate for a multiexponential analysis of a decay curve; t represents time and r has the physical meaning of a
relaxation rate. The data f(t) is a decay curve in the linearly sampled time domain, and the
desired inverse solution g(r) is a distribution of relaxation rates. Multiexponential fits are
notoriously difficult, and rarely extend beyond a superposition of just one or two terms; systematic misfit is often tolerated or accommodated by the use of ad hoc corrections (e.g.,
stretched exponentials). In this section an example of a broad asymmetric distribution of
relaxation rates simulated in the logarithmically sampled range of rmin = 10−3 to rmax = 100
will be used. This simulated g(r) is used to calculate f (t) according to Eq. 3; a random
normally distributed noise at 1% of the maximum f(t) intensity is added; and a pseudoinverse
solution is sought from the resulting noisy dataset. This is repeated throughout this section as
various aspects of regularization are examined.
1.2.1. Regularization Through Discretization
The first effect to consider is the discretization itself, already introduced in the previous
section. The act of choosing a good set of points in the parameter space for which to ask about
the value of the distribution function, by itself has a significant regularizing influence on the
inverse problem. The deterministic relationship between the time- and frequency-domain
point distribution is that a feature of FT is not available in a general inverse problem, and thus
the best strategy for selecting a proper level of discretization is not a hard-number prescription
similar to a Nyquist theorem, but a set of guidelines to follow:
• Select enough points in the grid of parameter values to allow for the entire physically relevant
range to be covered with sufficient resolution to reproduce all of the essential features in the data;
and no more.
• Acquire enough data points, sufficiently spread out in the observation domain to resolve contributions from different parameter values.
• Overdetermined problems, by at least a factor of two, are easier to solve.
• Higher parameter grid densities require better signal-to-noise ratio in the data.
Ultimately, numerical simulation and testing, and trial-and-error are required to establish a
good discretization scheme for a given problem. Near the optimum, the results of the calculation
should be largely independent of the exact choice of the discretization grids.
Figure 1 illustrates the regularizing effects of discretization alone, as applied to the example of a distribution of relaxation rates. Level of discretization is varied by changing the total
number of points across the parameter space. At just the right level of regularization (middle
plot) the recalculated g̃ is a faithful representation of the true g.
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107
Fig. 1. Regularization through proper discretization. Overly smooth pseudoinverse solution (top)
fails to reproduce some of the features of the true g, and insufficient regularization produces an unstable pseudoinverse (bottom). The optimum discretization (middle) faithfully reproduces the true g,
shown with a solid line in all three graphs.
1.2.2. The L-Curve
It is easy to recognize the optimal level of regularization when the true g(x) is known; in a
real experimental situation this is not the case. Fortunately, a convenient and very revealing
way to formally test any valid regularization mechanism is through the so-called L-curve,
which is a plot of the norm of the regularized solution g% vs the corresponding misfit
norm f − Kg% , as the regularization is varied (3,4). As the name implies, this curve has a characteristic L-shape illustrated in Fig. 2 for a number of grid densities in addition to the three
shown in Fig. 1. The corner of the curve represents the optimum regularization level, a point
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Fig. 2. L-curve obtained by systematically varying grid density. The three graphs of Fig. 1 correspond to three points on this curve, one near the corner (the optimum, middle plot of Fig. 1) and two
on the branches of the L. Upper-left region corresponds to loss of stability in g̃, and the lower-right
corner represents a loss of compatibility with the data.
of compromise between the two key quantities to be minimized if inverse solution is successful. Note how several points are crowded together near the corner of the L-curve; as mentioned before, near the optimum the exact choice of discretization becomes less important.
1.2.3. Regularization Through Singular-Value Decomposition Truncation
The second key to a successful pseudoinverse solution is to properly truncate the singularvalue decomposition (SVD) of the transformation matrix. Many discrete ill-posed problems
exhibit the property of a gradual decrease in the size of their singular values. The following
qualitative statement can be made for many physical problems: as the kernel matrix is
expanded in its singular values, the lower-valued ones tend to magnify the effects of noise in
the measured data. Thus, an effective way to regularize the inverse solution is to truncate the
SVD expansion, in a manner quite similar to the truncation of the Taylor series expansion of
a function. This has to be done in the region free of rapid changes in the size of the singular values. Yet again, there is no rigid prescription as to what is the appropriate level of truncation, and
multiple trial-and-error attempts might be required.
If SVD truncation is an appropriate regularization mechanism for a given problem, a systematic examination of the misfit norm again generates an L-curve, with the optimal truncation level
near the corner of it. This optimal level of SVD-truncation both provides numerical stability and
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Fig. 3. Regularization through SVD truncation. Overly smooth pseudoinverse solution (top) fails
to reproduce some of the features of the true g, and insufficient regularization produces an unstable
pseudoinverse (bottom). The optimum SVD truncation (middle) faithfully reproduces the true g,
shown with a solid line in all three graphs.
ensures compatibility with the data. Near the optimum, the inverse solution is stable with respect
to changes in the level of truncation. This is illustrated in Fig. 3 wherein, as before, the three
graphs illustrate the effects of (top to bottom): excessive, optimal, and insufficient regularization
through SVD truncation. The full L-curve is similar to Fig. 2 and is not shown.
1.2.4. Tikhonov Regularization
Whereas the previous two methods of regularizing a pseudoinverse arise from general
aspects of signal sampling, the true key to a successful inverse solution lies in the incorporation
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of additional physical constraints through Tikhonov regularization (5). The procedure is to minimize a modified misfit functional, with the least-squares term ensuring compatibility with
the data as usual (see Eq. 2), and the additional terms enforcing some essential property of
the resulting inverse solution:
Ψ{g} = f ( y) −
∫ g( x )K ( x , y)dx
2
+ λT {g}
(4)
where
2
⎧
g ( Tikhonov)
⎪⎪
2
T {g} = ⎨
g″ ( Philips)
⎪
⎪⎩− ∫ g log g dx ( maximum entropy)
(5)
For each form of regularization functional (three common ones are shown),∗ the dimensionless parameter λ controls the balance between compatibility with the data and the regularizing effects of T{g}. Clearly, the right choice for λ is crucial: if λ value is too small, a
stable solution will not be found; if λ is too large, essential physical features in the solution
will be obscured. It should be noted that other forms of the regularization functional are possible and for any given problem and finding the best one may represent a significant challenge;
the three popular ones shown in Eq. 5 are conceptually simple, have been well-studied
theoretically, and are known to work for a wide variety of physical data containing random
noise. Figure 4 illustrates the dependence of the inverse on the value of λ; in the interests of
brevity, the corresponding L-curve is not shown.
1.2.5. Example: Distributions of Relaxation Rates
As an illustration of a successful regularization of a difficult problem, consider the previously
introduced example of multiexponential analysis of a decay curve. The inherent difficulty of
fitting a noisy signal measured on a linear scale with an exponentially varying function means
that it is extremely difficult to answer even the most basic questions, for example, whether
the “spectrum” of relaxation rates is a broad continuous distribution or a sum of several
narrow processes, each with its own relaxation rate. Traditionally, each of the possibilities
would be modeled separately, and the fits compared with the data. Choosing the right kind of
a model function would be an essential qualitative step, left to the discretion of the investigator.
Regularizing the inverse solution allows us instead to remap the data into a distribution of
relaxation rates, in a model-free manner. Examination of the resulting distribution immediately
reveals the underlying physical mechanisms and the nature of model that should be pursued,
as shown in Fig. 5.
Two simulated relaxation rate distribution functions, ga(r) and gb(r), are used to generate two
time-domain signals according to Eq. 3; 1% random noise is added, and the resulting fa(t) and
fb(t) are then inverted over the grid of 100 logarithmically distributed relaxation rate values. The
two time-domain signals appear similar and do not provide clues to the nature of the underlying relaxation-rate distribution. On the other hand, mapping the data into the domain of
∗The
umbrella name of Tikhonov regularization is often used to refer to the calculational scheme defined by
Eqs. 4 and 5, and covers all forms of the regularization functional.
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Fig. 4. Tikhonov regularization. Overly smooth pseudoinverse solution (top) fails to reproduce
some of the features of the true g, and insufficient regularization produces an unstable pseudoinverse
(bottom). The optimum choice of λ (middle) faithfully reproduces the true g, shown with a solid line in
all three graphs.
relaxation rates through a regularized inverse solution immediately differentiates between a
“broad asymmetric hump” in ga(r) and a “superposition of three distinct lines” in gb(r). To be
sure, there are discrepancies between the true and the recalculated distribution functions;
these arise as a consequence of the noise, and can only be reduced by improving the quality
of the input data. Nevertheless, the essential qualitative nature of the underlying processes is
clearly revealed. This is a simulated example; however, the same method was used to establish that photochromic reaction spiropyran ↔ merocyanine occurs not in a distribution of
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Fig. 5. Example of a successful inverse solution: multiexponential decay. Solid lines in the insert
represent true distributions of relaxation rates; discrete points are the inverse solutions over a grid of
100 logarithmically spaced relaxation rates.
local environments with a wide spectrum of site-dependent local reaction rates, but as a coexistence of several different isomeric reactions, which run in parallel (6).
1.2.6. Self-Consistency
L-curve is an excellent graphical tool for reviewing effects of regularization. With the help of
an L-curve, proper choice of discretization grids and level of SVD truncation can be established
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113
fairly quickly for many cases of interest. However, explicit optimization of the regularization
parameter λ may require additional numerical work, and the method of choice for a broad
range of experimental situations is the so-called self-consistency method (7,8) that allows the
optimal value for λ to be determined solely from the data itself. The numerical details are
beyond the scope of this discussion, but the essence of the method is to monitor not just the
total misfit norm, but the statistical distribution of the misfit function, and to insist that this
distribution is identical to the distribution of experimental noise, determined independently,
typically from a baseline area of the experimental data wherein no known signal exists or
from a separate null experiment. Conceptually simple, this idea is of fundamental importance; it allows one to state that the obtained approximate solution reflects all of the information from the measured data, without overfitting the noise. In this sense, the pseudoinverse
becomes not just one possible solution, but the best solution within the constraints of the
experimental errors, and can be thought of as a faithful mapping of the experimental data into
the parameter space.
2. Case Studies
2.1. De-Pakeing: Distribution Functions in Biomembrane NMR
Anisotropy of molecular motions in biological and model membranes is responsible for
partial motional averaging of the orientation-dependent second-rank tensor interactions, such
as anisotropic chemical shift, nuclear dipole–dipole, and nuclear quadrupolar interactions.
Axial symmetry of fast reorientational motions produce the following fundamental scaling
relationship
ω(θ) = xP2 (cos θ) = x
3 cos 2 θ − 1
2
(6)
between the spectroscopic observable ω and the inherent motionally averaged anisotropy
parameter x. Although both have the dimensions of frequency, it is important to note that
only ω domain is accessible experimentally. Here θ is the angle between the symmetry axis
of fast molecular motions and the external magnetic field. The system may exhibit more than
a single inherent time-averaged anisotropy, giving rise to an anisotropy distribution function
g(x), for example, the order parameter profile SCD(n) (where n labels different molecular
sites along the hydrocarbon chain) extracted from the quadrupolar splittings of an 2H NMR
spectrum; a set of isotropic chemical shifts or chemical shift anisotropies from a 13C NMR
spectrum, and so on.
On the other hand, experimental spectra may contain contributions from domains of
different orientations, giving rise to an orientational distribution function p(θ). Individual
domains remain static on the time-scale of the experiment, but each anisotropy x makes a
spectral contribution associated with every orientation present in the sample. This gives rise to
a continuous lineshape function, sx(ω), for each x. Because P2(cos θ) varies from +1 to −1/2,
each sx(ω) contributes to the total S(ω) in the range from x to −x/2. The other contributing
function, p(θ), is the probability of encountering a domain oriented at an angle θ with respect
to the external magnetic field, for example, for a single crystal at 0°, p(θ) = δ(θ) and thus
sx(ω) = δ (ω − x); for a perfectly random powder p(θ) = sin θ, and from Eq. 6 it immediately
follows that sx(ω) = [3x(x + 2ω)]−1/2.
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Fig. 6. Both orientational and anisotropy distribution functions, p(θ) and g(x), contribute to the formation of the powder spectrum, as described by Eqs. 7–8. Here, p(θ) = sin θ is used as appropriate
for a random distribution of domain orientations.
Both g(x) and p(θ) matter, and the spectral lineshape S(ω) can be written in two equivalent ways:
⎡
∂θ ⎤
⎡
∂y ⎤
S (ω) =
∫ g( x )⎢⎣ p(θ) ∂ω ⎥⎦ dx
=
∫ p(θ)⎢⎣g( x ) ∂ω ⎥⎦dθ
(7)
(8)
with Eq. 7 implying a g(x)-weighted sum of lineshapes, and Eq. 8 implying a p(θ)–weighted
sum of oriented spectra. Figure 6 illustrates the relationship between g(x), p(θ), and S(ω).
Inverse Algorithms in NMR of Biomembranes
115
Clearly, Eqs. 7 and 8 describe an inverse FIE problem, where either of the two distribution
functions g(x) and p(θ) need to be extracted from the experimentally measured spectrum S(ω)
that now plays the role of f (y) from Eq. 1. The case of S(ω) → g(x) (Eq. 7) implies that the form
of p(θ) is known, for example a fully random distribution of orientations is p(θ) = sin θ. This is
the so-called de-Pake-ing∗ (9–11) that is widely used for the analysis of solid-state NMR
spectra. For example, “de-Pake-ing” 2H NMR spectra of chain-deuterated phospholipids in a
model membrane represent directly the orientational-order parameter profile SCD(n) that
reports on the molecular motions in the bulk of the membrane. Monitoring changes in this
order parameter profile caused by structural phase changes, interactions with other membrane
constituents, and so on, is a sensitive and powerful experimental tool (12).
The case of Eq. 8 wherein the spectral anisotropies g(x) are known and the orientational
distribution p(θ) needs to be measured is a separate inverse FIE problem. It has been used,
for example, to examine the aligning properties of a porous media filled with a liquid crystalline material (13). However, in biomembrane NMR, de-Pakeing is the inverse problem of
primary interest. Unfortunately, in this case one needs to know what p(θ) is, and it is well
known that the assumption of p(θ) = sin θ is often inadequate, because of the effects of partial alignment of membrane bilayers by an external magnetic field.
2.2. Magnetic Alignment: Extracting Both g(x) and p( θ)
Lipid bilayers have anisotropic magnetic susceptibility,
∆χ = χ − χ ⊥ < 0
(9)
which gives rise to an interaction between an induced
magnetic moment of each rsmall
r
domain of area A, thickness d, and bilayer normal n, with the external magnetic field H ; the
resulting torque
r
r r r r
(10)
τ = ∆χ A d (H · n )( H × n )
r r
orients the domain preferentially so that n ⊥ H . Physically, this corresponds to spherical vesicles being deformed into ellipsoids with their long axes preferentially aligned along the magnetic field. Figure 7 illustrates a typical example of a spectral distortion caused by such
partial magnetic alignment; the distortion is in the suppressed size of the spectral “shoulder”
associated with θ = 0 orientation, relative to the undistorted lineshape shown in Fig. 6. The
overall lineshape is dominated by the strong θ = 90° peak, and thus the distortion appears irrelevant. However, de-Pakeing is extremely sensitive to small lineshape changes, and the distribution of anisotropies it extracts is strongly affected by such distortions. This data was acquired at
a relatively low field strength of 7T, the effects are more pronounced at higher fields.
Therefore, it is essential to de-Pake in the presence of an unknown orientational distribution.
This is probably impossible for a completely arbitrary p(θ), as either anisotropy or orientation
may be responsible for a spectral contribution at any given observation frequency (see Eq. 6).
However, one can test and compare a parameterized family of models that describe a physically reasonable p(θ), de-Pake repeatedly, and select the best model parameters in the same
way, by finding a global minimum of the misfit norm. Fortunately, partial magnetic alignment
*After G. E. Pake who first reported lineshapes like that shown in Fig. 6.
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Fig. 7. Partial magnetic alignment of biomembrane domains by an external magnetic field is seen
in a 31P NMR spectrum of 83 wt% 1-palmitoyl-2-oleoyl-su-glycero-3-phosphoethanolamine (POPE) +
17 wt% 1-palmitoyl-2-oleoyl-su-glycero-3-phosphoglycerol (POPG) mixture in water at 25°C, in a
moderate field of 7T.
can be described as a relatively well-behaved continuous deformation of p(θ) from its random
form, sin θ. As a result, several one-parameter models for p(θ) are physically reasonable (14,15):
• Ellipsoidal (high correlation between orientations of adjacent domains):
pE (θ) ∝ sin θ ⎡⎣1 − (1 − κΕ ) cos 2 θ⎤⎦
−2
• Boltzmann (adjacent-domain orientations are uncorrelated):
pB (θ) ∝ sin θ exp ⎡⎣κΒ cos 2 θ⎤⎦
• Legendre polynomial expansion (general):
pL (θ) ∝ sin θ
∑
∝
i =1
Ai Pi (cos θ) ≈ sin θ(1 − Aκ L cos 2 θ)
when κE, κB, or κL, as appropriate, are varied, the orientational distribution function undergoes a
continuous deviation from the random p(θ) = sin θ. Use of these models for a simultaneous
extraction of g(x) and some limited information about the form of p(θ) requires a slight modification to the inverse procedure:
• The misfit function is modified to include the appropriate κ:
Ψ { g( x ); κ} = S (ω) −
∫ g( x )K (ω,x; κ)dx
2
+ λT { g( x )}
(11)
• To deal with the highly nonlinear dependence on κ, an appropriate range of κ-values is swept for
each model in turn and the misfit norm is monitored.
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Fig. 8. Simultaneous extraction of g̃(x) and p̃(θ) from a partially magnetically aligned powder spectrum: min Ψ → κ̃ → p̃(θ) → g̃(x). Dashed line is the same 31P NMR spectrum shown in Fig. 7,
whereas the solid line is g̃(x), the result of de-Pakeing using TR. The dotted line, visually indistinguishable from the input data, is the powder spectrum recalculated for the found g̃(x); the difference
between the two (the misfit) is shown below on a (×16) scale. The right insert shows that for the optimum value κE = 1.19 ± 0.005 the misfit norm is 100 times lower than for κE = 1, i.e., when a random
p(θ) = sin θ is assumed. The left insert shows the p(θ) that corresponds to this optimum κE; the random
sin θ is shown with a dotted line for comparison.
• The lowest of all minima in misfit norm corresponds to: (a) the best model and its optimal κ̃-value,
which in turn determines the best estimate p% (θ), and (b) the best inverse g% ( x ).
Figure 8 presents an example whereby a 31H NMR spectrum from a partially magnetically aligned model membrane sample was successfully de-Paked using Tikhonov regularization (TR), yielding not only the orientational order parameter profile, but also an estimate
of the degree of alignment, corresponding to an ellipsoidal-like deformation of a spherical
orientational distribution function. The optimal value κE = 1.19 has a simple physical meaning of the square of the ratio of the semiaxes of an ellipsoid of rotation, so in this case the
alignment is very mild. However, even this mild effect would obscure the emergence of the
second broad spectral component at about +15 KHz; this spectrum was measured near
a L α ↔ H II phase boundary, and this signal is from a very small amount of the second (HII)
component. Without the refinement that the inverse solution afforded herein, a small rise in
the spectrum, which is in fact, the peak of the HII powder pattern, would likely to have been
dismissed as an artifact. In magnetic fields typical of modern NMR spectrometers hydrated
biomembranes at biologically relevant temperatures exhibit a degree of alignment that
typically corresponds to κE ≈ 2 to 8.
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Fig. 9. Changes in the order parameter profiles of monounsaturated lipids with the double bond at
different positions along the chain, monitored through a small amount of d29-tetradecanol. SCD(n) are
extracted through de-Pakeing in the presence of partial magnetic alignment, separately for Lα and HII
phases, normalized by their respective SCD(1), and their ratio calculated. ∆6 DSPE: 1,2-dipetroseliuoylsn-glycero-3-phosphoethanolamine; ∆11 DVPE: 1,2-divacceiioyl-sn-glycero-3-phosphoethanolamine.
2.2.1. Chain Orientational Order Near Lα ↔ HII Transition
Being able to calculate g(x) in the presence of alignment means that spectral contributions
from different structural phases can be precisely measured, improving the quality of determining phase boundaries in complex structural mixtures (15). It is also essential for obtaining
orientational order parameters from the 2H NMR of chain-deuterated biomembrane samples.
TR-based de-Pakeing allows highly reliable SCD(n) determination (15,16), leading to a measurement of effects previously thought undetectable. For example, thermodynamic properties of
model membranes containing monounsaturated phospholipids depend strongly on the position of the double bond along the hydrocarbon chain, with very slight changes in chemical
structure causing dramatic shifts in phase-transition temperatures. To examine the chain orientational order for any anomalies associated with the position of the double bond a small amount
of deuterated tetradecanol can be added. It is known to localize in the bilayer and act as a
reliable reporter of the order parameter profiles (17), and this can be exploited to monitor subtle changes in the order parameter profiles as the sample undergoes L α ↔ H II structural phase
transition (18). The result is shown in Fig. 9 wherein a local modulation clearly shifts with the
position of the double bond. Plotted on the vertical axis is a dimensionless ratio of order parameters representing, in essence, the extent to which average local order remains “bilayer-like” as
the sample undergoes a dramatic change in local monolayer curvature associated with the
Inverse Algorithms in NMR of Biomembranes
119
formation of the HII phase. The effect is delicate, and comparable in magnitude with the
distortions in the order parameter profile caused by magnetic alignment; the ability to extract
g% ( x ) simultaneously with p% (θ) is essential in making it visible.
2.2.2. Alignment and Structural Organization
Figure 9 shows how to overcome a “distortion” resulting from a partial alignment; the
parameters of optimal p̃(θ) gathered in the process of obtaining an inverse solution yield additional information about the sample. However, it is interesting to note that an inverse solution
may be as instructive when it fails as when it succeeds. For example, one-parameter models
for orientational distribution functions may fail to produce a satisfactory inverse. This suggests presence of additional complexity in the sample that necessitates exploring higher-order
terms in the Legendre polynomial expansion, or some other combination of nonoverlapping
orientation-dependence functions. With some increase in processing power and extra care
taken in the acquisition of high-fidelity NMR data, one can study samples containing a superposition of multiple contributions from different structural arrangements, for example, a mixture of randomly oriented small micelles or multilamellar vesicles, and partially aligned giant
unilamellar vesicles or even strongly aligned bicelles. The search for a minimum of misfit
norm now extends over a two-parameter space: κ of the aligned fraction and the fraction of
total sample that is in the aligned phase. Figures 10 and 11 test the “golden standard” of
aligned biomembrane samples, namely, a glass-aligned “sandwich” preparation of a model
membrane system; d35-SDhPC* in this case. Sharp peaks typical of aligned samples tend to
obscure the broad powder patterns of the unaligned fraction and with some adjustment to the
phasing of the spectrum the latter may blend into the baseline. The resulting spectrum may
appear perfectly aligned, and de-Pakeing it—unnecessary. However, Fig. 10 shows an inverse
solution obtained through de-Pakeing and using a single-parameter ellipsoidal model. The
data is indistinguishable from the recalculated “fit” to the spectrum; however, a careful examination alerts us to a systematic misfit in the wings of the spectrum, making this inverse solution inadequate. On the other hand, the inverse shown in Fig. 11 is free of systematic misfit,
and a good de-Paked spectrum is obtained. This inverse solution is consistent with the sample containing a 70/30% (±2%) aligned/randomly oriented mixture of structural phases, with
the oriented one having the mean distribution of angles of about ±3°. This spectrum was
selected as being one of the best glass-aligned examples; “fully aligned” samples that in fact
contain 40% or more of unaligned phase are quite typical. More than a third of the sample
goes essentially unnoticed.
More than a diagnostic tool to monitor the quality of sample preparation, the structural phase
information obtained in the process is valuable by itself; for example, such two-parameter fits
have been applied to the spectra of a so-called bicellar short/long-chain lipid mixture
(1,2-dimyristoyl-sn-glycero-3-phosphocholine:1,2-dicaproyl-sn-glycero-3-phosphocholine in a
4.1:1 molar ratio). Such spectra, similar in appearance to the one shown in Figs. 10 and 11 have
been interpreted as evidence of “perfect” alignment in bicelles (19). The information obtained
using mixed-model de-Pakeing, allows construction of a rich, structural-phase diagram of a
bicellar mixture (15); throughout most of the temperature range bicelles turn out to be mixed
structural phases. This has been confirmed independently in a recent detailed study (20).
*1-steroyl-2-docosahexaenoyl-sn-glycero-3-phosphocholine.
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Fig. 10. NMR spectrum of a glass-aligned bilayer sample of d35-SDhPC, de-Paked using a singleparameter ellipsoidal model. Notation and the arrangement of inserts is identical to Fig. 8. The data
(dotted line) and the fit (dashed lines) are essentially indistinguishable, but a systematic misfit in the
wings of the spectrum reveals the inadequacy of this model.
Figures 10 and 11 demonstrate the need for high-fidelity experimental data and the care
that must be used in the data processing: the spectral features that yield such detailed information are small and easily obscured by excessive filtering or the artifacts of other common
processing methods that are used to improve the visual appearance of the spectra. These figures also illustrate how an improved, albeit computationally costly, data analysis leads to a
realization of a need for a better model, to an introduction and use of such a model, and
finally, to a better and more complete understanding of the story told by the data. As mentioned earlier, the model itself becomes a “fit parameter,” and is improved on in the process.
This is the true strength of regularization.
2.3. Regularization in Two Dimensions
The challenge in the examples presented here lies primarily in the quality of data acquisition; if a reliable data set is available and the kernel function is reasonably well behaved, the
inverse is likely to succeed. Most frequently, the data is one-dimensional (1D), although some
successful inverse solutions have been performed to obtain 2D distribution functions as well.
For example, 2D DOQSY and DECODER data obtained from spider dragline silk was
inverted to yield a Ramachandran map, i.e., a 2D distribution of backbone torsion angles
(φ,ψ) (21,22). Tikhonov regularization functional (Eq. 5) does not explicitly depend on the
(possibly multidimensional) parameter-space variable x, and so the minimization problem
Inverse Algorithms in NMR of Biomembranes
121
Fig. 11. NMR spectrum of a glass-aligned bilayer sample of d35-SDhPE, de-Paked using a 70/30%
aligned/randomly oriented mixed model. Notation and the arrangement of inserts is identical to Fig. 10.
The minimum in Ψ(κ) is relatively shallow, especially seen on the logarithmic scale, partly because
of a small systematic misfit near zero associated with an isotropic line, likely from residual HDO that
has a different isotropic chemical shift. This artifact does not influence the de-Paked spectrum of the
lipid. Note the absence of a systematic misfit in the wings of the spectrum, as in Fig. 10. A large value
of the optimal κE = 220 ± 10 corresponds to a narrow (about ±3°) distribution of the bilayer normals
in the aligned fraction (the left insert).
can be reformulated as an equivalent 1D problem. Reindexing {gij } → {gl }, where i = l ... m,
j = l ... n, and l = m × (j − 1) + i, converts a 2D grid of (φi,ψj) values into an equivalent 1D
distribution (22). However, even this problem, although massive in scope, belongs to the
same class as all of the examples in the preceding sections, namely, that of the overdetermined problems: the size of the dataset exceeds that of the parameter space.
Somewhat unexpectedly, this is not a necessary condition, although for underdetermined
problems the difficulties of obtaining a successful inverse solution become greater, and the
importance of choosing the best regularization functional for a given problem increases. In
2D NMR problems, in particular, there is considerable redundancy in the data, as the adjacent rows and columns of the 2D dataset typically overlap in their dependence on the adjacent values in the parameter distributions. Regularization terms that provide a certain amount
of 2D coupling in the parameter space can help to overcome the difficulties of inverting
underdetermined problems. One such example is shown in Fig. 12, wherein a distribution of
backbone torsional angles g(φ,ψ) is simulated over a 5° × 5° grid, with two Gaussian peaks
representing a certain mixture of near α-helical and near β-sheet motifs, a total of 2701 points
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Fig. 12. Intensities of 190 CP MAS cross-peaks are calculated from a simulated 73 × 37 Ramachandran
torsional angle distribution g(φ, ψ) (on the left) and 0.1% random noise added. The resulting dataset is
inverted using Tikhonov regularization with Laplacian as the regularization functional; the result (on the
right) faithfully reproduces the initial distribution function (29). Inversion using only the least-squares
minimization fails to recover the true g(φ, ψ), at any noise level (not shown).
in parameter space. A set of 190 1H–13C CP MAS cross-peak intensities is then calculated
from this distribution, as appropriate for Alzheimer’s β-amyloid fibrils (23). A 0.1% random
noise is added to the simulated
data, and this vastly underdetermined (190-by-2701) inverse
2
functional, T ( g ) = ∇ 2 , g , another well known form of regularization. The true distribution is
successfully recovered (Fig. 12); for an experimentally obtained data this would allow a
determination of the relative amounts of different backbone motifs in the sample. The question of whether Alzheimer’s fibrils pack in a parallel or an antiparallel fashion is of significant structural importance; this can now be tested using a fairly limited dataset that can be
obtained indirectly in CP MAS and finite-pulse radio-frequency-driven recoupling at constant time experiments (24). Without regularization only very limited qualitative fits can be
performed; these suggest the presence of a fairly broad distribution of backbone angle pairs
(25,26).
3. A Regularization Primer
The computational “cost” of getting some first-hand experience with regularized inverse
solutions is surprisingly low. In addition to many excellent implementations in the form of C
and Fortran libraries (e.g., GENEREG* [27]), high-level macro language implementations have
been gaining popularity (e.g., Regularization Tools for MatLab† [28]). The latter-style tools
*www.fmf.uni-freiburg.de/service/sg_wissinfo/Software/.
†www2.imm.dtu.dk/~pch/Regutools/.
Inverse Algorithms in NMR of Biomembranes
123
Fig. 13. Example of SciLab code implementing inversion through Tikhonov regularization, as
appropriate for the multiexponential analysis example shown in Fig. 5.
are particularly interesting because they allow a novice user to focus on the algorithm rather
than on the computational details, as they provide a compact feature-rich vector-oriented
syntax. The example shown in Fig. 13 was used to solve the multiexponential example of the
Introduction.
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For clarity, this sample code uses a particular value for all relevant parameters (grid density,
level of SVD truncation, regularization parameter λ); by adding appropriate loops, all figures
of the Introduction can be obtained. The heart of the code is a very short function defined at
the beginning; this function is called from the main routine, which contains mostly housekeeping calls. The only computational task in the main routine is to fill in the values of the
discretized kernel matrix, a sum of exponential decay curves in this case; this portion of the
code needs to be changed to use it for another inverse problem. The code is written in the syntax of SciLab*, a public-domain alternative to MatLab†; using it with MatLab involves only
minor syntactic changes.
Acknowledgments
Heartfelt thanks to Hartmut Schäfer for many illuminating discussions and for his help
with software; Robert Tycko for the use of simulated CP MAS data; and Ivan Polozov for the
glass-aligned d35-SDhPC spectrum. Figures were prepared using the software developed at
the Tri-University Meson Facility, Vancouver, Canada. This work was supported by Natural
Sciences and Engineering Research Council of Canada.
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