Reconstruction Algorithms for Next-Generation Imaging:Multi-Tiered Iterative Phasing for Fluctuation X-ray Scattering andSingle-Particle Diffraction
Jeffrey J. Donatelli
Mathematics Department, Lawrence Berkeley National Laboratory
Center for Advanced Mathematics for Energy Research Applications (CAMERA)

Emerging Imaging Techniques Enabled by XFELs
Fluctuation X-ray Scattering (FXS)

DiffractionPattern
Pattern
Scattering

Single-Particle Diffraction (SPD)Particle Stream

X-raysDiffraction Pattern

Outline of Talk

1) Fluctuation X-ray Scattering (FXS)i) Description of experimentii) Mathematical formulation of the FXS reconstruction problemiii) Multi-tiered iterative phasing (M-TIP) for FXS reconstructioniv) Experimental challengesv) Reconstructions from experimental data

2) Single-Particle Diffraction (SPD)i) Description of experimentii) Single-particle M-TIP for SPD reconstructioniii) Results

Fluctuation X-ray Scattering (FXS)

I

Several X-ray diffraction images are collected from particles in solution using X-ray
exposures below rotational diffusion times

I

Multiple particles per shot

I

Images contain angular fluctuations:I

Angular correlations are computed from each image J (n) and then averaged:C(q, q 0 , ∆φ) =

I

≠

1
2π

Z 2π
0

∑

J (n) (q, φ + ∆φ)J (n) (q 0 , φ)dφnMedium to high resolution structure can be determined from the correlation data

Kam Z (1977) Determination of macromolecular structure in solution by spatial correlation of scattering fluctuations. Macromolecules 10(5):927–934
Kam Z, Koch MH, Bordas J (1981) Fluctuation x-ray scattering from biological particles in frozen solution by using synchrotron radiation. Proc Natl Acad
Sci USA 78(6):3559–3562

SAXS vs. FXS
Small Angle X-ray Scattering (SAXS)Particles rotate during imaging:

Fluctuation X-ray Scattering (FXS)
Particles are frozen in place during imaging:

Full rotational integration:

Finite rotational averaging:

Image =Z

IR dR

Image =

SO(3)

M
X

I Rj

j=1

Images are angularly isotropic:

Images contain angular fluctuations:

Can only extract angular average: (J(q, φ) = Image)Can extract angular correlations: (J(q, φ) = Image)

SAXS(q) =Z 2π0J(q, φ)dφ0

C(q, q , ∆φ) =Z 2π0J(q, φ)J(q 0 , φ + ∆φ)dφOutline of Talk

1) Fluctuation X-ray Scattering (FXS)i) Description of experimentii) Mathematical formulation of the FXS reconstruction problemiii) Multi-tiered iterative phasing (M-TIP) for FXS reconstructioniv) Experimental challengesv) Reconstructions from experimental data

2) Single-Particle Diffraction (SPD)i) Description of experimentii) Single-particle M-TIP for SPD reconstructioniii) ResultsBasic Diffraction Theory and Notation
Coordinate Notation:

CartesianrqReal space
Fourier space

Polar(r, θ, φ)(q, θ, φ)Electron Density: ρ(r) (Gray isosurface)

Structure Factors:ρ̂(q) =

Z

ρ(r)e−2πiq·r dr

R3

Intensity Function: Diffraction measures information about the intensity function
I(q) = |ρ̂(q)|2
Goal: Determine information about the electron density ρ from a set of diffraction images

Diffraction Images

Intensity Function I(q)Electron Density ρ(r)Spherical Harmonics - “Fourier Series on a Sphere”
Ylm
0
1
l
2
3
−3

−2

Spherical Harmonic Expansion:
I(q, θ, φ) =2

3

Ilm (q)Ylm (θ, φ)I(q, θ, φ)Ylm∗ (θ, φ) sin θdθdφIl(1−l) (q)...

0Il(−l) (q1 )Il(−l) (q2 )Il = 
.
.
.
Il(−l) (qN )

∞
l
X
X

1

Z 2π Z π

Ilm (q) =Matrix Format:
0m

l=0 m=−lSpherical Harmonic Transform:

Vector Format:
Il (q) = Il(−l) (q)−1Il(1−l) (q1 )Il(1−l) (q2 ).
.
.
Il(1−l) (qN )

0

...
...
.
.
.
...

T

Il0 (q)...

Il(l−1) (q)Ill (q)Il0 (q1 )Il0 (q2 ).
.
.
Il0 (qN )

...
...
.
.
.
...

Il(l−1) (q1 )Il(l−1) (q2 ).
.
.
Il(l−1) (qN )Ill (q1 )Ill (q2 ).
.
.
Ill (qN )Relation between FXS and the 3D Intensity Function
Average Correlation Function: (Images: J (1) , . . . , J (Ndp ) )C(q, q 0 , ∆φ) =

≠

1
2π

Z 2π

J (n) (q, φ + ∆φ)J (n) (q 0 , φ)dφ0∑nLegendre Polynomial Expansion: (Ndp → ∞, uniformly random orientations)
1 X
Pl cos θ(q) cos θ(q 0 ) + sin θ(q) sin θ(q 0 ) cos ∆φ Bl (q, q 0 ),4π∞C(q, q 0 , ∆φ) =l=0

where θ(q) = arccos(qλ/2), and, up to a set of scaling factors

Bl (q, q 0 ) =lX∗
Ilm (q)Ilm(q 0 )

m=−l= < Il (q), Il (q 0 ) >Legendre Polynomials

Features:
I

Uncorrelated noise effects (e.g. shot noise) vanish when averaging the correlations over a
sufficient number of images (apart from a sharp “noise peak” at q = q 0 ,∆φ = 0)

I

For M particles per shot, B0 ∼ M 2 and Bl ∼ M for l > 0

FXS Reconstruction Problem - The Phase Problems
In order to determine structure from correlation data, two tiers of phase problems must be solved:
1) Classical Phase Problem (I(q) = |ρ̂(q)|2 )Scalar Phase Problem
ρ̂(q) =

p

I(q) eiφ(q)The complex phases φ(q) need to be determined in order to recover ρ.
2) Hyperphase Problema) Autocorrelation Data: (Bl (q) := Bl (q, q), Bl (q) = ||Il (q)||2 )Il (q) =p

Vector Phase Problem

Bl (q) ul (q), where u∗l (q)ul (q) = 1

The phase vectors ul (q) need to be determined in order to recover I.
b) Cross-Correlation Data: (Bl (q, q 0 ), Bl = Il Il∗ )√√Rank 2l + 1 Eigenvalue Decomposition: Bl = Vl Λl Vl∗ = (Vl Λl )(Vl Λl )∗Matrix Phase Problem

Il = VlpΛl Ul , where Ul∗ Ul = I2l+1

The phase matrices Ul need to be determined in order to recover I.

FXS Reconstruction Problem - Summary
Relation between structure and FXS data:
Fourier
Transform

ElectronDensity

ρ(r)

Z

ρ(r)e

−2πiq·r

SquareStructure
Modulus
Factors
dr

ρ̂(q)

2

|ρ̂(q)|

Spherical Harmonic
Transform

Intensity
FunctionI(q)Z 2πZ π00Intensity Spherical
Harmonic CoefficientsIlm (q)I(q, θ, φ)Ylm∗ (θ, φ) sin θ dθdφComputeFXS DatalX∗
Ilm (q)Ilm(q 0 )FXS Data

Bl (q, q 0 )m=−l

FXS Reconstruction Problem:
Determine the electron density ρ given the Bl (q, q 0 ) quantities along with additional constraints.
ElectronDensityρ(r)Classical Phase
Inverse Fourier
Structure
Intensity
ProblemTransform
Factors
Function
Z
p
I(q)
ρ̂(q)
ρ̂(q)e2πiq·r dq
I(q)eiφ(q)Inverse Spherical
Harmonic Transform∞ X
lX

Intensity Spherical
Harmonic CoefficientsIlm (q)Ylm (θ, φ)Ilm (q)l=0 m=−lHyper-phase
Problemp

Bl (q) ul (q)pVl Λl UlFXS Data

Bl (q, q 0 )I

Complex phases φ(q) need to be recovered (classical phase problem)

I

Phase vectors ul (q) or phase matrices Ul need to be recovered (hyper-phase problem)

I

Additional constraints on the solution are needed to make the problem well-posed (e.g.
constraints on size, positivity, symmetry, density statistics)

Outline of Talk

1) Fluctuation X-ray Scattering (FXS)i) Description of experimentii) Mathematical formulation of the FXS reconstruction problemiii) Multi-tiered iterative phasing (M-TIP) for FXS reconstructioniv) Experimental challengesv) Reconstructions from experimental data

2) Single-Particle Diffraction (SPD)i) Description of experimentii) Single-particle M-TIP for SPD reconstructioniii) Results

Overview of Multi-Tiered Iterative Phasing (M-TIP) for FXS Reconstruction1) Develop a set of Bregman projection operators, each of which projects a model estimate to
the closest object that satisfies a given constraint.
Constraint
Model
Model Constraints:
I

Consistency with the Bl data

I

Compact support

I

Nonnegativity

I

SymmetryI

Similarity to another structure

I

Statistical properties

Projected Model

2) Starting with a random initial model, iteratively apply projections to enforce constraints until
the projected models are consistent with all constraints and data.

Real-space
Projections

Correlation
Projections

Fourier-space
Projections

Real-Space Projectors
I

Support Projector: To enforce a support region Sß(PS ρ)(r) =

ρ(r),
0,

if r ∈ S,
otherwise.

PS ( )=
Density

SupportI

Nonnegativity: P+ ρ = max(ρ, 0)I

Upper Bound: Pτ ρ = min(ρ, τ )ICombinations: PS+ , PSτ , PS+τI

Symmetry Projector: To enforce symmetry given by a point group G ⊆ O(3)(PG ρ)(r) =1 X
ρ(Or).
|G|O∈G

PG ( ) = + + + =Fourier-Space Projectors
I

Magnitude Projector: Projects density ρ to satisfy I = |ρ̂|2
ρ̂(q)

◊
(P
M (I)ρ)(q) =
I

p

I(q)

(PM (I)ρ̂)(q)ρ̂(q)|ρ̂(q)|p
I(q)Autocorrelation Projector: Projects intensity I to satisfy Bl (q) = ||Il (q)||2
Il (q)(PA I)l (q) =Ip

Bl (q)(PAI)l (q)Il (q)||Il (q)||p
Bl (q)

Cross-Correlation Projector: Projects intensity I to satisfy Bl = Il Il∗(PC I)l = Vl
v2
v3

p

Λ l Ul ,Ul =arg min ||D(Il − Vl
Ul ∈U(2l+1)

Il (q3)p(Bl = Vl Λl Vl∗ )Λl Ul )||FIl (q1)v1
Il (q2)Vl√

Λl Row Vectors

Il Row Vectors

Unitary Alignment(PC I)lMulti-Tiered Iterative Phasing (M-TIP)
Fourier-Space Operator:
F ρ = PM (PA |ρ̂|2 )ρ or PM (PC |ρ̂|2 )ρ.

Real-Space Projector:
PS∗
Error Reducing (ER) Scheme:

ρ(n+1) (r) = (PS∗ F ρ(n) )(r).
Hybrid Input-Output (HIO) Scheme:
ρ(n+1) (r) =ßc F ρ(n) )(r) = 0,(F ρ(n) )(r),if (PS∗(n)c(n)ρ (r) − β(PS∗ F ρ )(r), otherwise,

where P c ρ = ρ − P ρ and 0 < β ≤ 1.
Fourier
Transform

ρ(r)Real-SpaceConstraints⊗| · |2

ρ̂(q)

I(q)

Spherical
Harmonic Transform

Real-Space
Projections

PS , PS+ ,PSτ , PS+τ ,PG

Inverse Fourier
Transform new

ρnew (r)ρ̂Phases⊗⊗Correlation Projection

PA , PC(q)

Ilm (q)I new (q)FXS Data

Bl (q, q 0 )Inverse Spherical
Harmonic Transform new
Ilm (q)Magnitude Projection
PM

Donatelli JJ, Zwart PH, Sethian JA (2015) Iterative phasing for fluctuation X-ray scattering. Proc Nat Acad Sci USA 112(33):10286–10291

Reconstructions of pLGIC from Simulated FXS Data
Original

Reconstruction from FXS
Cross-Correlation data
Bl (q, q 0 ), l ≤ 20

Reconstruction from FXS
Cross-Correlation data
Bl (q, q 0 ), l ≤ 20
5-fold symmetry constraint

Outline of Talk

1) Fluctuation X-ray Scattering (FXS)i) Description of experimentii) Mathematical formulation of the FXS reconstruction problemiii) Multi-tiered iterative phasing (M-TIP) for FXS reconstructioniv) Experimental challengesv) Reconstructions from experimental data

2) Single-Particle Diffraction (SPD)i) Description of experimentii) Single-particle M-TIP for SPD reconstructioniii) Results

Systematic Issues in Experimental FXS Data
BeforeSolvent Background

BeforeCommon Mode

Jet/injector streaks
After

After

Line offsets
BeforeAfter

Beam Center Refinement

How to remove systematic issues and noise:
I Correct for issues directly in each image
I Remove effects by subtracting out correlations of average of all images and blank shots:
Cfiltered = Chits − Caverageimage − Cmisses
Model remaining issues during the reconstruction process (Noise projectors)
I Filtering via sparse representations
INoise Projector
Instead of fitting directly to the noisy observed data Dobs , we fit to a noise-filtered version which
is updated during each iteration via
PN Dmod = Dmod + ∆D,
modwhere Dis simulated by solving the forward problem with the current model and ∆D solves
min ||∆D|| subject to

1
Ndata


X (Dmod + ∆D)i − Diobs 2
i

σi2

< 1.

Example: 2D Phase Retrieval from Noisy Intensity Measurements

Iterative Phasing
without Noise Projector

Original Model

Diffraction Pattern with
Gaussian Noise

Clean Diffraction Pattern

Iterative Phasing
with Noise Projector

Noise Projector for Correlations
Variances σi2 can be estimated via symmetry violation about ∆φ = π/2, π, 3π/2!

Clean Correlations

Noisy Correlations

M-TIP can be modified to model noise by applying the noise projector in an additional tier.
SHT

FT

Hyperphases

Phases

Real-SpaceConstraints

IFT

IALT

ISHT

FXS Data

ALT

Correlation Filtering via Low-Rank Decompositions
The correlation function can be expressed in terms of a small number of basis elements:
Low-Rank Legendre Decomposition: (lmax ≈ πqmax D, D = particle size)

X

lmax

C(q, q 0 , ∆φ) =Pl (cos θ(q) cos θ(q 0 ) + sin θ(q) sin θ(q 0 ) cos ∆φ)Bl (q, q 0 )l=0

Low-Rank Spherical-Bessel Decomposition: (Kl ≈ qmax D − l/π)
Bl (q, q 0 ) =

Kl
Kl
X
X

blk1 (q)blk2 (q 0 )cl (k1 , k2 ),k1 =1 k2 =1

where
blk (q) =ul,kjl+1 (ul,k )jl (2πqD)
u2l,k − (2πqD)2

Low-Rank Eigenvalue Decomposition (2l+1 nonzero eigenvalues)

X

2l+1

cl (k1 , k2 ) =λli vli (k1 )vli (k2 ),λl1 , . . . , λl2l+1 ≥ 0

i=1Enforcing this low-rank expansion allows us to drastically filter the correlation data!
I Can be performed by combining the noise projector, regularized linear inversion, andprincipal component analysis

Correlation Filtering via Low-Rank Decompositions - Example
Simulated Data SizeI

Nq = 320, N∆φ = 400

I

Nq2 × N∆φ = 4 × 107 total measurementsI

4 × 102 degrees of freedomI

105 data redundancy

Correlation
Filtering1
Noisy SNR = 500

Filtered SNR = 200 (105 improvement!)Outline of Talk

1) Fluctuation X-ray Scattering (FXS)i) Description of experimentii) Mathematical formulation of the FXS reconstruction problemiii) Multi-tiered iterative phasing (M-TIP) for FXS reconstructioniv) Experimental challengesv) Reconstructions from experimental data

2) Single-Particle Diffraction (SPD)i) Description of experimentii) Single-particle M-TIP for SPD reconstructioniii) Results

Experimental FXS Data from CroV and PBCV (Back Panel Only)
CroV Experimental DataI

1-5 particles per shotI

30K images

I

50 nm resolution at
detector edge

PBCV Experimental DataI

50-100 particles per shotI

60K images

I

50 nm resolution at
detector edge

Credit: Proposal Team - Kerfeld, Schlichting, Ourmazd, Fromme & Zwart teams

Data Processing and Structure Solution - Donatelli, Malmerberg & Zwart

Reconstructions from Experimental Autocorrelation Data (Back Panel Only)
CroV Reconstruction (l ≤ 12, no symmetry enforced)

Isodensity surface

2D slice

Bl (q) fits

PBCV Reconstruction (l = 0, 6, icosahedral symmetry enforced)

Isodensity surface

2D slice

Bl (q) fits

Reconstruction of PBCV from Experimental FXS Data (With Front Panel, 12 nm data)

EM Data Bank Structure

Reconstruction from Experimental
Autocorrelation Data
(icosahedral symmetry enforced)

Experimental Single-Particle Diffraction Data from RDV and PR772
RDV Experimental DataI

1 particle per shotI

332 images

I

12 nm resolution at
detector edge

PR772 Experimental DataI

1 particle per shotI

566 images

I

12 nm resolution at
detector edge

Credit: Proposal Team - Single Particle Initiative

Data Processing and Structure Solution - Kurta, Donatelli, Yoon, & Zwart

Reconstructions from Experimental Cross-Correlation Data (No symmetry enforced)
RDV

Resolution: FSC - 13.5 nm, PRTF - 17.7 nm

PR772

Resolution: FSC - 12.6 nm, PRTF - 16.9 nm
Kurta RP, Donatelli JJ, Yoon CH, Pande K, Bielecki J, Svenda M, Xavier PL, Fromme P, Aquila A, Zwart PH, Mancuso AP (2017) Correlations inscattered X-ray laser pulses reveal nanoscale structural features of viruses (In Review)

Outline of Talk

1) Fluctuation X-ray Scattering (FXS)i) Description of experimentii) Mathematical formulation of the FXS reconstruction problemiii) Multi-tiered iterative phasing (M-TIP) for FXS reconstructioniv) Experimental challengesv) Reconstructions from experimental data

2) Single-Particle Diffraction (SPD)i) Description of experimentii) Single-particle M-TIP for SPD reconstructioniii) Results

Single-Particle Diffraction (SPD)Particle Stream

X-raysDiffraction Pattern

I

Diffraction patterns are collected from individual molecules, i.e., one particle per shotI

Each particle is delivered to the beam at random and unknown orientations through a liquid
droplet or aerosol delivery system

Solem JC, Baldwin GC (1982) Microholography of living organisms. Science 218:229–235.
Neutze R, Wouts R, van der Spoel D, Weckert E, Hajdu J (2000) Potential for biomolecular imaging with femtosecond x-ray pulses. Nature 406:752.

SPD Reconstruction Problem
Single-Particle Diffraction Images:
Image J (k) samples I along a spherical slice
rotated according to the image orientation
Rk :J (k) (q, φ) = I (Rk ) (q, θ(q), φ),where θ(q) = arccos(qλ/2).
Challenges:
1) Orientation Problem: Determine the orientation Rk of each image J (k) .
2) Intensity Reconstruction: Extract the 3D intensity function I from the set of images.
3) Classical Phase Problem: Reconstruct the electron density ρ from the intensity function I.

1)

2)

3)

Outline of Talk

1) Fluctuation X-ray Scattering (FXS)i) Description of experimentii) Mathematical formulation of the FXS reconstruction problemiii) Multi-tiered iterative phasing (M-TIP) for FXS reconstructioniv) Experimental challengesv) Reconstructions from experimental data

2) Single-Particle Diffraction (SPD)i) Description of experimentii) Single-particle M-TIP for SPD reconstructioniii) Results

M-TIP for Single-Particle Diffraction (SPD)SPD Operators:
1) Orientation Matching: Aligns each image to the current
3D intensity model

2) Intensity Synthesis: Projects the current 3D intensity
model to be as consistent as possible with the set of 2D
images at their estimated orientations

Fourier
Transform

ρ(r)Real-SpaceConstraints

⊗

I(q)

Phases

Real-Space
Projections

Inverse Fourier
Transform

ρnew (r)| · |2ρ̂(q)ρ̂new (q)⊗+Spherical
Harmonic TransformIlm (q)Orientation MatchingandIntensity Synthesis

I new (q)Inverse Spherical
Harmonic Transform⊗SPD Data

J (k)(q, φ)

new
Ilm
(q)

Magnitude Projection
Donatelli JJ, Sethian JA, Zwart PH (2017) Reconstruction from limited single-particle diffraction data via simultaneous determination of state,orientation, intensity, and phase. Proc Nat Acad Sci USA 114(28):7222–7227.

Outline of Talk

1) Fluctuation X-ray Scattering (FXS)i) Description of experimentii) Mathematical formulation of the FXS reconstruction problemiii) Multi-tiered iterative phasing (M-TIP) for FXS reconstructioniv) Experimental challengesv) Reconstructions from experimental data

2) Single-Particle Diffraction (SPD)i) Description of experimentii) Single-particle M-TIP for SPD reconstructioniii) Results

Reconstruction from Simulated SPD Data with Single-Particle M-TIP
Original Model of pRb: (5.5 Å)Reconstructions from simulated clean SPD data:

24 images
5.8 Å

12 images
6.3 Å

6 images
7.5 Å

Reconstruction from Mixed-State SPD Data with Single-Particle M-TIP
Sialic acid binding protein (SiaP) in its open and closed states:

Original Models (5.5 Å):

Reconstructions from simulated mixed-state SPD data: (48 images, 24 from each state)

5.8 Å

5.6 Å

Reconstruction from Noisy SPD Data with SP M-TIP + Noise Projector
Images with Simulated Shot Noise (5.5 Å, 8000 photons/image, .25 photons/Shannon-angle at edge)

5
4
3
Shot Noise2
1
Clean Image

Noisy Image# photons

Single-State Reconstruction

Multi-State Reconstructions

192 images
6.4Å

384 images (192 images per state)
6.3 Å
6.8 ÅSingle-Image Reconstructions of RDV and PR772 with SP M-TIP
RDV

SP M-TIP with
Icosahedral constraint

PR772

SP M-TIP with
Icosahedral constraint

Summary
Emerging techniques enabled by XFELs:
1) Fluctuation X-ray Scattering (FXS) - Measure angular correlations from the diffraction
patterns of particles in solution
I
I
I

Allows for multiple particles per shot (100% hit rates)
Does not require knowledge of particle orientations
Robust to uncorrelated noise2) Single Particle Diffraction (SDP) - Collect diffraction images from single molecules
I
I

Reveals structural information about individual molecules
More information per image compared to FXSMulti-tiered iterative phasing (M-TIP):
I

General iterative reconstruction framework for simultaneously determining all degrees of
freedom

I

Able to boost the effective information content of the system via enforcement of real-spaceconstraints during each iterationI

Can determine 3D structure from FXS without enforcing symmetryI

Can determine 3D structure from SPD from a sparse set of images

Future Work:I

Model experimental error and uncertainty as part of the M-TIP reconstruction process

I

Scale up single-particle M-TIP code to use more RDV & PR772 data and model moreheterogeneity

I

Extend M-TIP to other experiments

Acknowledgements
Methods Development/Data Processing:
I Peter Zwart (LBNL)
I Kanupriya Pande (LBNL)I Erik Malmerberg (AstraZeneca)
I Ruslan Kurta (European XFEL)I James Sethian (LBNL & UC Berkeley)
I Nicholas Sauter (LBNL)
I Billy Poon (LBNL)
I Aaron Brewster (LBNL)
I Chuck Yoon (SLAC)I Adrian Mancuso (European XFEL)
Data Collection:
I L605 Proposal teamI

Kerfeld team (MSU)
Schlichting team (MPI)
Fromme team (ASU)
I Ourmazd team (UWM)
I Zwart team (LBNL)
I
I

I

Single Particle InitiativeComputation of Integral Transforms
Polar Grid:I
Inradial nodes: rn = N, qn = Qn
N

inclination angle nodes: θl0 = arccos(xl0 ),where xl0 are Gaussian quadrature nodes
with weights wl00I

azimuth angle nodes: φm0 = 2πm
M

Discrete Spherical Harmonic Transform: (FFT + Gaussian quadrature)ρlm (rn ) =1 XX
ρ(rn , θl0 , φm0 )Ylm∗ (θl0 , φm0 )wl0 .
M
l0

m0

Polar Fourier Transform: Computed by applying the spherical Hankel transformlρ̂lm (qn ) = 4π(−i)

Z ∞ρlm (r)jl (2πqn r)r2 dr,0using a sine/cosine series approximation and then evaluating the spherical harmonic seriesρ̂(qn , θl0 , φm0 ) =

XX
l

m

ρ̂lm (qn )Ylm (θl0 , φm0 ).

Interparticle Coherence Effects
Complication - FXS theory assumes that the scattering from multiple
particles add incoherently, but XFELs have a fully coherent beam!
I

Assume M > 1 particles per shotI

Aj (x) - complex scattering from the j-th particle at pixel x
Coherent Sum:

Incoherent Sum:
Imageincoh (x) =

M
X
j=1

|Aj (x)|2

Imagecoh (x) =

M
X
j=1

2

Aj (x)

Interparticle Coherence Effects (cont’d)Relation between correlations from a coherent and incoherent sum: (d = beam×sample density)
Ccoh (q1 , q2 ) = Cincoh (q1 , q2 )ˆ 2 )|2 + a2 |d(q
ˆ 1 )|2 + a3 |d(q
ˆ 1 − q2 )|2 + a4 |d(q
ˆ 1 + q2 )|2+ a1 |d(qˆ 1 − q2 )d(−q
ˆ
ˆ
ˆ
ˆ
ˆ
+ a5 <[d(q1 )d(q2 )] + a6 <[d(q1 + q2 )d(−q1 )d(−q2 )]ˆ 1 )|2 |d(qˆ 2 )|2+ a7 |d(qFor a sufficiently large and smeared-out sample volume and beam, dˆ ≈ δ (Dirac Delta function):
Ccoh (q1 , q2 ) = Cincoh (q1 , q2 ) + b1 δ(q1 ) + b2 δ(q2 ) + b3 δ(q1 − q2 ) + b4 δ(q1 + q2 )In polar coordinates:
Ccoh (q, q 0 , ∆φ) = Cincoh (q, q 0 , ∆φ) + c1 δ(q) + c2 δ(q 0 ) + c3 δ(q − q 0 )δ(∆φ) + c4 δ(q − q 0 )δ(∆φ − π)

Coherence effects are concentrated in peaks at q or q 0 = 0 and q = q 0 for∆φ = 0 or π, which can be masked out!

Low q

High qComparison of FXS Reconstruction Methods
Serial Approach: Solve the hyperphase problem and then solve the classical phase problem
I In general, insufficient number ofIntensity
Density
Constraints
Constraints
constraints on the intensity to determine
the hyperphases
I

Can be performed for icosahedral, helical,
or cylindrical particle symmetry [1-5]

Data

Solve hyperphaseproblem

Solve classicalphase problem

Density

Black-box Optimization: Find density that is consistent with data via black-box optimization [6]I Leverages density constraints to determine
Density
Constraints
hyperphases (No symmetry requirement)
I

Unable to exploit structure of the problem

I

Poor convergence properties - can require
week of computing hours to determine
low-resolution structure

Data

DensityMulti-Tiered Iterative Phasing: Iteratively update estimates for the density, scalar phases, andhyperphases via projections [7]I

Leverages density constraints to determine
hyperphases (No symmetry requirement)

I

Fully exploits structure of the problem toincrease accuracy and accelerate
convergence

I

Fast - medium to high resolution
reconstructions require 5-30 minutes on asingle CPU core

Data

Density
Constraints

Updatedensity

Updatescalar phases

Updatehyperphases[1]Saldin et al (2011) [2]Poon et al (2013) [3]Poon and Saldin (2015) [4]Wibowo and Saldin (2016) [5]Uddin (2016) [6]Liu et al (2013) [7]Donatelli et al (2015)

Comparison of SPD Reconstruction Algorithms
Current Approaches to SPD Reconstruction:
I Manifold embedding [1-6]
I Common curve analysis [7-10]
I Expectation maximization (EMC) [11-13]
The above approaches are serial in nature - they solve the problem in two separate steps:
1) Orientation determination and intensity reconstruction2) Phasing - to determine the electron density from the reconstructed intensity function

Single-Particle
ImagesIntensity
Constraints

Density
Constraints

Determine orientations
and 3D intensity

Solve classicalphase problem

Density

Single-Particle Multi-Tiered Iterative Phasing (M-TIP) [14]I

Iteratively updates estimates for thedensity, phases, orientations, and intensity

I

Can leverage real-space density constraints
to help determine orientations and boost
information content of the system(Reduces number of required images)

Density
Constraints

Updatedensity

Single-Particle
ImagesUpdateOrientations[1] Fung et al (2009) [2] Schwander et al (2010) [3] Moths and Ourmazd (2011) [4] Giannakis, Schwander, and Ourmazd (2012)[5] Schwander et al (2012) [6] Hosseinizadeh et al (2014) [7] Shneerson (2008) [8] Yang, Wang, and Marchesini (2010)[9] Bortel and Tegze (2011) [10] Yefanov and Vartanyants (2013) [11] Loh and Elser (2009) [12] Tegze and Bortel (2012)[13] Flamant et al (2016) [14] Donatell, Sethian, and Zwart (2017)

Update3D Intensity