Noname manuscript No. [626441]

Noname manuscript No.
(will be inserted by the editor)
Non-reciprocal Light-harvesting Antennae
Julian Juhi-Lian Ting
February 2, 2017
Abstract From a point of view of classical electrodynamics, the performance of
two-dimensional shape-simplied antennae is discussed based upon the shape of
naturally designed systems to harvest light. The modular design of nature is found
to make the antenna non-reciprocal, hence more ecient. We further explain the
reason that the light harvester must be a ring instead of a ball, the function of the
notch at the LH1-RC complex and the non-heme iron at the reaction center from
a physical point of view, and comment about how our prediction might be veried
experimentally.
1 Introduction
Photosynthesis that occurs daily in bacteria, algae and plants has been a subject
traditionally studied by (bio)chemists[1,2]. A major task in this eld is trying to
understand the molecular structures. Because of the inadequate instrumental reso-
lution, chemists struggled for decades to know the structure of the light-harvesting
antenna. Without precise structures, people tried to guess the content within the
black box. The method used to study the paths and time scales of transfer of
excitation energy is, in general, spectrometry.
The Fenna-Matthews-Olson (FMO) complex of green sulfur bacteria was the
rst pigment-protein complex to have its structure analyzed with x-ray diraction
[3]. These bacteria live deep in the sediments of water in which they encounter
only few photons of light per day; they hence require an ecient mechanism to
harvest light. They consist of chlorosomes and the FMO complex that function
together. The chlorosomes are the largest known photosynthetic complexes in ex-
istence, comprising about 250,000 pigment molecules, whereas antenna complexes
of other types consist of only a few dozen pigments. The FMO complex mediates
the transfer of excitation energy from light-harvesting chromosomes to the bac-
terial reaction center (RC) embedded in a membrane with a trimeric structure
(symmetry C3). Each of the three monomers contains seven bacteriochlorophyll a
molecules.
De-Font Institute, Taichung 40344, Taiwan, R.O.C.

2 Julian Juhi-Lian Ting
In 1984, the rst crystal structure of RC was determined, which resulted in a
Nobel Prize four years later[4].
Beginning about 1995, scientists have acquired a reasonably complete pic-
ture of the bacterial light-harvesting (LH) system [5{7]. Both the inner antenna,
LH1, and the outer antenna, LH2, are assembled from the same modules to form
rings. Each module consists of two short -helical polypeptides coordinating one
carotenoid and three bacteriochlorophylls. The exact numbers of modules involved
for both complexes are variable [5,8,9]. LH2 is smaller and consists of nine units
forRhodopseudomonas acidophila [10,11] with inner diameter 36 A and outer di-
ameter 68 A; LH1 is larger, as it contains the RC, and is composed of 16 units, for
Rhodospirillum rubrum with outer diameter 116 A and central diameter 68 A [8].
There are two further subtleties in the structure of LH1 that are important.
First, the ring has a mysterious opening[12]. Some earlier authors simply draw a
misleading cartoon which propagated across succeeding generations[13,1]. Second,
the RC contained in the LH1 has a non-heme iron.
Many such structures have been subsequently analyzed[14], some of which we
list in Table 1. For only Rhodopseudomonas palustris is the X-ray crystal structure
known for its LH1-RC complex to date. Symmetries are notably unaltered even
under strained condition for LH3, i.e., a variant of LH2[15].
Following a knowledge of the structure, a question remains. Why has God
made a light harvester from modules to form a tambourine-like shape instead of
a spherical shape or a serpentine shape, for instance? The number of modules
involved is apparently unimportant.
The standard explanations for photosynthesis are given in terms of chemistry[1,
2]. We calculated a simplied LH1 model based upon chemical rate equations
for the shape acquired, but advanced no farther than others[22]. Some physicists
have recently been working on photosynthesis because of a question posed by
Lee et al.[23]. They considered the remarkable eciency of transfer to be a kind of
quantum search, but a consensus is a kind of quantum random walk[24,25]. Such a
random-walk interpretation of light-harvesting is, however, not new[26]. Although
there is a dispute whether photosynthesis is a quantum process or a classical
process, their work marked a shift of interest to the mechanism responsible for the
energy transfer.
In the present work LH are considered from a classical electrodynamic point of
view, as the light-harvesting antennae function the same as radio antennae from
many perspectives except their size and their reception frequencies. We try to
explain photosynthesis from a physical point of view. Our interest is the structural
factor for a light harvester, and have arrived at a useful interpretation. We explain
also the function of the notch at the LH1-RC complex still unknown to biochemists,
and the function of the non-heme iron at the reaction center.
2 Loop Antennae
Two simplied shapes are shown in the gure. Figure 1 (a) is simple and can
be solved with algebraic methods, whereas gure 1 (b), although resembling LH1
more closely, has a resonance frequency only slightly modied from that of gure
1 (a). We must hence consider only gure 1 (a). The notch is essential for LH1

Non-reciprocal Light-harvesting Antennae 3
protein PDB ID symmetry cartoon
LH1-RC from Rhodopseudomonas palustris [16] 1PYH
LH2 B800-850 from Rhodopseudomonas acidophila [11] 1NKZ C9
LH2 B800-850 from Rhodospirillum molischianum [17] 1LGH C8
peridinin-chlorophyll from Amphidinium carterae [18] 1PPR C3
FMO from Prosthecochloris Aestuarii [19] 3EOJ C3
FMO from Pelodictyon phaeum [20] 3VDI C3
allophycocyanin from cyanobacterium Phormidium [21] 4RMP C3
Table 1 Various light-harvesters with structural symmetry. PDB ID is the protein ID assigned
by Protein Data Bank http://www.rcsb.org/pdb/
as the received energy must be taken from some point. The Rhodopseudomonas
palustris molecule of 1PYH shown in the table also has a notch.
In engineering, antennae of such shapes are well studied: they are called loop
antennae and loop antennae with line feed, respectively[27]. Figure 1 (a) was used
by Heinrich Hertz when he rst demonstrated radio waves in 1888; the only dier-
ence is that the wavelength of our consideration is small, into the optical region.
Loop antennae divide into two categories depending upon their size relative to
the wavelength of operation. If an antenna has a radius smaller than the wavelength

4 Julian Juhi-Lian Ting
a=9.2nm
a=9.2nm
Fig. 1 Antennae of two simplied shapes. According to experimental data, the radius is the
average of 68 A and 116 A .
of operation, it is called a small-loop antenna; otherwise it is called a resonant-loop
antenna. As the wavelength, , of operation of LH1 or LH2, 800 900nm, is much
larger than the radius of the antenna, 9 :2nm, the light-harvesting antenna is a
small-loop antenna, which has a small eciency and serves mainly as a receiving
antenna.
To arrive at the electromagnetic properties of such an antenna there are a
complete way and a simple way. We begin with the complete way.
Let the radius of the loop located at the origin be a, and the plane of the
loop bexy; let the angle from the xaxis be. If current Iaround the loop is
uniform and in phase, the only component of the vector potential is A, as shown
in Figure 2 (a). The innitesimal value of Aat a point away from the loop by
(a)
(b)
Fig. 2 The coordinate system uses.

Non-reciprocal Light-harvesting Antennae 5
distancercaused by two diametrically opposed innitesimal dipoles is
dA=dM
4r; (1)
in which dM= 2j[I]acos[sin(2acossin=)] d,is the angle relative to the
vertical axis through the center of the loop, and [ I] =I0expfj![t(r=c)]gis the
retarded current on the loop with I0being its maximum value. After integration
we obtain
A=j[I]a
2rJ1(2asin
); (2)
in whichJ1is a Bessel function of rst order.
As the source of sunlight is remote, we consider the far-eld eects. The far
electric eld of the loop has only a -component E=j!A that is in the plane
of the loop. Therefore,
E=1202a[I]
rJ1(2asin
): (3)
The corresponding magnetic eld in free space is
H=a[I]
rJ1(2asin
): (4)
The second method is to decrease the innite number of dipoles used in the
preceding method into four short linear dipoles as follows.
Let the area of the antenna be A, which is commonly called the aperture of
the antenna, and the length of the dipoles be d, as shown in Figure 2 (b). Hence
d2=a2A (5)
The far electric eld is
E=1202[I] sin
rA
2: (6)
The termA=2is a pure ratio: it is the aperture in terms of wavelength. The
magnetic eld is obtained on dividing the intrinsic impedance of the medium, i.e.
H=E
120=[I] sin
rA
2(7)
in vacuum.
Eq. (6) is a special case of Eq. (3), just as Eq. (7) is a special case of Eq. (4),
as for small arguments of the rst-order Bessel function J1(x)x=2.
The (radiation or receiving) resistance at the loop terminals can be obtained
from
P=I2
0
2R (8)
in whichI0is the maximum current on the loop and Ris the resistance. The total
powerPis obtainable on integrating the Poynting vector
S=1
2jHj2ReZ (9)
over a large sphere, in which Zis the impedance of the medium. The resistance
thus obtained for a small-loop antenna is proportional to 1 =4.
Antennae are generally constructed of size about =4. As the size decreases,
the terminal impedance (complex resistance) becomes increasingly reactive, which
hinders the transfer of power; the coupling between the circuit and the wave be-
comes unsatisfactory.

6 Julian Juhi-Lian Ting
3 Non-reciprocal Antennae
A regular antenna both receives and emits, because Maxwell's equations are sym-
metric with respect to time reversal[28], but a light harvester that functions solely
as a receiver must receive much better than it emits. What we seek is more than
an optical equivalent of an electronic rectier, which is articial, but an optical
counterpart of a duckbill check valve in
uid dynamics. Such a valve is a naturally
designed passive device that exists in a human heart. In particular, these atri-
oventricular valves include the mitral valve and the tricuspid valve, and semilunar
valves.
As no modal or polarization properties of sunlight are known, we require mech-
anisms to break the (Lorentz) time-reversal symmetry[29]. There are several pos-
sibilities to make such an optical check valve:
{Faraday rotator[30,31];
{circulator[28,32];
{duplexer[33{35];
{nonlinearity [36{41].
All four mechanisms are well known to the scientic and engineering communities.
{The rst one, a Faraday rotator, requires an externally applied magnetic eld.
All LH1-RC complexes are equipped with a non-heme iron at the RC, i.e the
Rhodobacter sphaeroides RC[42], which could provide the required eld. The
iron is not bound to any protein and can be exchanged with zinc ( Zn2+) or
manganese ( Mn2+), just like a pearl in a mussel's mouth that chemists call
"coordinated"[43{45]. The role played by the non-heme iron has long been
questioned, if not unknown to biochemists[46]. Biologists know that it serves
as a source or sink of electrons during electron transfer or redox chemistry,
without realizing the implication of magnetic elds[47]. The fact that the non-
heme iron can be exchanged conrms our prediction about the role played by
the iron. Presumably, it can also be exchanged with other magnetic atoms.
The eect is, however, typically small and decreases proportionally to 2
in an organic material, although there are isolated reports of large Faraday
rotation[48,49]. In contrast, a nano-scale (quantum) optical isolator/circulator
that requires neither great intensity of light nor a strong magnetic eld but
relies upon polarization has been built [50,32].
Furthermore, although the main concern of the present paper is about a bacte-
rial light harvester, molecular data show that photosystems of algae and plants
likely evolved from the photosystems of green-sulfur bacteria; there are hence
many analogous functions and similar structures. The photosystems (I and
II) of algae and plants are typically equipped with non-heme iron and even
manganese ( Mn)[51,52].
{The geometry of the second one, circulator, resembles our LH well. The FMO
complex is a 3-port circulator, Rhodopseudomonas acidophila LH2 (PDB ID
1NKZ and 1KZU) is a 9-port circulator, whereas LH1 for Rhodospirillum
rubrum is a 16-port circulator. Such optical circulators are commonly used
in ber-optic transmission to direct the optical signal from one port to another
port and in one direction only.
{Duplexer (or multiplexer) is the terminology used in communication engineer-
ing, whereas physicists call them time-dependent material, which might func-

Non-reciprocal Light-harvesting Antennae 7
tion together with the circulator. Biochemists know that when a chlorophyll
pigment absorbs light, it loses an electron to the RC, which might also mean
duplexing [2]. Interpreted in terms of mechanics, the photon received might
cause the-helical polypeptides/ carotenoid/ bacteriochlorophyll complex to
alter its shape or orientation, hence rendering impracticable the
ow of the
electromagnetic wave in the other direction.
A recent nding of photon anti-bunching from LH supports such a scenario,
although no further reason for the physics behind conformational change is
provided[53,54]. Furthermore, the method used is
uorescence spectrometry
with which conformational change is impossible to observe directly[55,56]. Cur-
rent methods of imaging with a microscope are not only limited by the wave-
length used but require harsh conditions or direct mechanical interaction with
the samples, hence becoming inappropriate for live cells, which is the reason
that traditionally only spectrometry has been used. A newly invented imaging
method using phonons (acoustic waves) of sub-optical wavelength or photonic
crystal-enhanced microscope might be able to observe the route that a photon
takes on entering and leaving LH, to see how the LH alters its conformation[57,
58]. Otherwise, we might be able to fabricate a nano-scale optical isolator, by
chemical synthesis for example, according to scaling laws, to assess whether
it generates spectra of the same characteristics as the real light harvester[59].
The molecule thus synthesized need not be organic, or even be light-sensitive,
as what we are considering is the geometrical eect. Computer simulation can
also provided some information.
{Fourthly, the LH are apparently nonlinear media. The internal quantum states
of the non-heme iron at the RC can serve to control the direction of the light
propagation[32].
The above statement has not yet excluded a spherical shape, which occurs on
evoking the Poincar e-Brouwer theorem[60]. This theorem was originally proven
by Poincar e and is sometimes called the hairy-ball theorem; it states that there
exists at least one point on the surface of a sphere at which vectors of electric and
magnetic elds become equal to zero. The Poynting vector at this point is also
zero. This theorem indicates a toroidal shape instead of a spherical circulator.
4 Discussion
There is dispute whether the ring should be considered as a conductor or not.
What we show is those seemingly erroneous assumptions can indeed reach some
important conclusions. Presumably, the light-harvesting process up to the special
pair of RC can be explained all in terms of classical electrodynamics. After the
special pair, since the light (wave) has transferred into an electrons, we must
consider quantum mechanics, for which reason a special pair is a dimer instead of
a trimer. The special pair resembles two optical bers of a lately realized nano-
circulator[32].
There is some subtlety in the structures about whether the LH comprise over-
lapping rings or non-overlapping rings. The vigilant reader will nd that we con-
sider them as overlapping in the case of small-loop antennae but non-overlapping
when discussing non-reciprocity. Simplication gives insight, which is the key to
our discovery and which is a major approach of physicists[61].

8 Julian Juhi-Lian Ting
There are indeed systems with more sophisticated symmetries, i.e. symme-
try D3 of red-shifted phycobiliprotein complexes isolated from the chlorophyll f-
containing cyanobacterium Halomicronema hongdechloris[62]; C-Phycocyanin from
Leptolyngbya[63], and C-phycoerythrin from cyanobacterium Phormidium[64]. Sym-
metry D3 represents a twisted or a stereo symmetry C3. Furthermore, the number
of bacteriochlorophyll in each module might be another level of symmetry: rst, it
means self-similarity as discussed in fractal theories[65], second, the number three
instead of two, like the special pair in the RC, might indicate classical instead of
quantum.
In summary, in this work, we sought another possibility to consider a light-
harvesting antenna as a device to receive electromagnetic waves. We nd that a
modular LH is an eective design to make an antenna receive more than it emits.
Furthermore, we provide an explanation for the function of the notch and the
non-heme iron at the LH1-RC complex, unknown at the present to the scientic
community. We show that there are both physics and chemistry inside the problem
of photosynthesis. All four categories of non-reciprocity deserve further investiga-
tion. As the geometry is known, we interpret a mechanism that might be useful for
the future production of articial light-harvesting systems and should be added to
the list of design principles[66].
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