The Blackbody Radiation-like Equation (BRE) that fits Data from Protein Folding, Single-Molecule Enzyme Catalysis, Whole-Cell RNA Metabolism, and T-Cell Receptor Diversification
a square on the y the time or the conditions under which the RNA levels are measured.
Reproduced from [3].
1
Sungchul Ji, Ph. D.
INTRODUCTION
Department of Pharmacology and Toxicology, Ernest Mario School of Pharmacy, Rutgers University, Piscataway, N.J.
[1] The blackbody radiation equation discovered by M. Planck in 1900 is shown in Figure 1.
[2] One of the main purposes of this poster is to present the evidence that the blackbody radiation-like equation (BRE) (see
Eq. (3) below) fits experimental data obtained from:
(A)
(B)
(C)
(D)
(E)
[email protected]
4
Fitting Protein Stability Data from E. coli to the Blackbody
Radiation-like Equation (BRE)
y = (a/(Ax + B)5)/(eb/(Ax + B) - 1)
with a = 4.1x1013, b = 365; A = 8.5 and B = 9
4,300 proteins; data from K. Dill et al., PNAS 108:17876-82 (2011)
6
(6) As shown in Figure 11, BRE also fits the distributions of generation probabilities of the
human T-cell receptor CDR3 sequences predicted by Murugan et al. based on a statistical
model of the T-cell receptor gene recombination processes [2].
(7) Murugan et al. [2] predicted two distributions of CDR3 generation probabilities , one for
the naïve non-productive and the other for the naïve productive repertoires (see Figure 12), of
which the distribution of the productive sequences is simulated in Figure 11. The deviation of
the BRE curve from the Murugan et al’s “productive repertoire” curve is pronounced toward
the tail end. This deviation should be significantly reduced if the distribution of the
nonproductive sequence repertoire is used. This is encouraging because Murugan et al’s
theoretical distribution model is constructed to account for the redceptor gene variaons of the
non-productive repertoire and not that of the does not apply to the generation probabilities of
the productive receptor gene sequences [2, p. 7].
400
[3] Another purpose of this poster is to propose a possible explanation for these surprising findings in terms of a model of
enzyme catalysis that shares some mechanistic similarities with laser (see Figure 15).
350
Number of Proteins
300
(C)
BRE
250
200
Empirical
150
100
50
0
0
10
20
30
40
50
60
ΔG, in units of RT
(b)
5
Figure 12. Generation probabilities of all the
CDRs sequences in the naïve and productive
repertoires computed using an inferred distribution
[2, p. 13]. The productive repertoire curve was
reversed in its x-direction by subtracting the xvalues from a constant before plotting in Figure
11.
Figure 11. The fitting of the T-cell receptor
CDR3 gene sequence variations to BRE.
Figure 1. (a) Blackbody radiation: All matter emits light with different wavelengths and intensities when heated
producing the so-called blackbody radiation spectra shown in (b). (c) The blackbody radiation equation
discovered by M. Planck (1858-1947) in 1900.
Blackbody radiation equation: u(λ, T) =
(8πhc/λ5)/(ehc/λkT
Blackbody radiation-like equation (BRE): y =
(a/x5)/(eb/x
y =(a(Ax +
– 1)
Figure 13. Recombinases viewed as
molecular machines that are designed to edit
DNA nucleotide sequences to generate new
sequences. CTD = C-terminal domains; NTDs
= N-terminal domains
– 1)
(3)
Figure 7. A microarray experiment involves the following 6 key steps: (1) Isolation of RNAs from broken
cells. (2) Synthesis of fluorescently or radioactively labeled cDNAs (called “targets”) from isolated RNAs
using reverse transcriptase and appropriately fluorescently labeled nucleotides. (3) Preparing a microarray
using either EST (expressed sequence tag, i.e., sequences several hundred nucleotides long that are
complementary to the stretches of the genome encoding RNAs) or oligonucleotides (synthesized right on the
microarray surface) (called “probes”). (4) Hybridizing the fluorescently labeled cDNAs (targets) with the
ESTs covalently bound to the microarray surface (probes). (5) Measuring the signals of the targets hybridized
to the probes on the surface of the microarray using a computer-assisted microscope. (6) Displaying the target
signals as a table of numbers, each registering the signal intensity of a square on the microarray surface which
is proportional to the concentration of the targets (and ultimately to the RNA levels in cells before isolation)
located at row x and column y, row indicating the identity of the genes encoding the targets, and y the time or
the conditions under which the RNA levels are measured. Reproduced from [6a].
2
3
Figure 3 The measurement of the turnover of a cholesterol
oxidase (COx) molecule in the presence of cholesterol (0.20
mM) and oxygen (0.25 mM). FAD, is fluorescent when in its
oxidized state with an average relative intensity of about 130
units (which is referred to as the “on“ state) and nonfluorescent when in its reduced state with an average intensity
of about 40 units (which is referred to as the “off” state). The
length of “on” state is also called the “waiting time”, the time
an enzyme molecule awaits before it catalyzes a chemical
reaction. Reproduced from
http://www.nigms.nih.gov/News/Reports/single_molecules.ht
m.
Figure 8. Typical results of microarray measurements of
the time-dependent genome-wide RNA levels (also called
RNA trajectories) in budding yeast . The RNA trajectories
were measured at 0, 4, 120, 360, 450 and 850 minutes
after replacing glucose with galactose. TL = transcript
level. Data from [5].
Cell Wall Biognesis (9 RNAs)
140
120
100
TL
Figure 2. The fluorescence image of single
molecules of cholesterol oxidase (COx)
immobilized in agarose gel. When FAD is
illuminated at 442 nm, the prosthetic group, FAD,
emits fluorescence at 530. Each individual
fluorescent spot indicates the presence of a single
molecule of COx. The intensity variations are due
to different longitudinal positions of COx molecules
in the gel. Reproduced from
http://www.nigms.nih.gov/News/Reports/single_mol
ecules.htm
1. Ground state
of an enzyme
80
60
Rate = k [enzyme-substrate complex] = k [enzyme][substrate]
0
-200
0
200
400
600
800
1000
Time, minutes
Figure 9. The tree steps involved in analyzing
RNA kinetic data in terms of the blackbody
radiation-like equation (BRE). (a) The original
RNA kinetic data in the form of RNA trajectories.
(b) Each RNA trajectories from (a) can be
represented s a point in the 6-dimensional RNA
concentration space. (c) The distances (or dissimilarities) among all possible pairs of RNA
trajectories in (a) can be calculated from (b) using
the Euclidean formula and these n(n-1)/2 distances
can be classified into a set of distance classes to
generate a histogram.
f(w), measured & caclauted
20
15
10
5
0
0
50
100
150
200
250
300
-5
Waiting time, w , ms
Figure 4. The distribution of the kinetic parameters (ontimes) measured from COx molecules. (Left) The
distribution of the on-times (i.e., the duration of the times that
the FAD molecule remains in its oxidized state and
fluorescent). This phenomenon of varying on-times is known
as ‘dynamic disorder’ or ‘dynamic heterogeneity’. (Right)
The distribution of the rate constants, k2, for the reduction of
FAD to FADH2 measured from 33 COx molecules. At any
given time, the k2 values measured from different molecules
of COx vary by a factor of up to 5. Reproduced from
http://www.nigms.nih.gov/News/Reports/single_molecules.ht
m
Figure 5. Single-molecule enzymic activity of
cholesterol oxidase activity measured in [7] (see blue
squares) is partially simulated by BRE (see purple
squares).
Figure 10. The fitting of the RNA distance data
(see Figure 3 (c)) into BRE, Eq. (3), with a = 109,
b = 47.5, A = 2 and B = 2 [6, 7]. Circles are
experimental data, and the solid curve is derived
from BRE. The curve fitting software was
written by Kenneth So.
TL
Glycolysis Pathway (18 RNA molecules)
3
E
Blackbody Radiation
Enzymic Catalysis
E5
C‡
E4
Cn
E3
Ci
E2
E
C3
E1
C2
E0
C1
Figure 6. A comparison between blackbody radiation and enzymic
catalysis. (Left) Blackbody radiation involves promoting the energy levels
(vibrational, electronic, or vibronic) of oscillators from their ground state E0
to higher energy levels, E1 through E5. The wavelength of the radiation (or
quantum) absorbed or emitted is given by ΔE = Ei – E0 = hf, where Ei is the ith
excited-state energy level, h is the Planck constant, f is the frequency, and ΔE
is the energy absorbed when an oscillator is excited from its ground state to
the ith energy level. Alternatively, blackbody radiation can be thought of as
resulting from the transitions of electrons from one energy level to another
within matter, e.g., from E1 to E0, from E2 to E0, etc. (Right) A single
molecule of cholesterol oxidase (COx) is postulated to exist in n different
conformational states (i.e., conformational substates of Frauenfelder et al. [8])
denoted here as Ci with i = 1 to n. Each conformational state (or conformer)
is thought to carry a set of sequence-specific conformational strains, or
conformons, and can be excited to a common transition state (denoted as C‡)
by thermal fluctuations, leading to catalysis [1, Section 12.12].
Time, min
(4) The histogram (Figure 10) of the frequency vs. distance classes of the RNA trajectories belonging to a
given metabolic pathway can be fit into a blackbody radiation-like equation (BRE), Eq. (3), that is similar
in form to the Planck’s radiation equation, Eq. (2) [6].
(5) BRE has also been found to fit single-molecule kinetic data of cholesterol oxidase [4, 7] and protein
stability data [2].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
where A is the “pre-exponential factor”, ΔG‡ is the activation Gibbs free energy. The activation free energy ΔG‡
can be expressed as
ΔG‡ = G‡ - Gi
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)
where G‡ is the Gibbs free energy level of the enzyme at the transition state and Gi is the Gibbs free energy level of
the ith ground state.
Inserting Eq. (3) into Eq. (2) leads to
7
k = Ae-ΔG‡ /RT
= Ae- (G‡ - Gi )/RT
= A(e- G‡/RT)(eGi/RT )
= A’eGi/RT
(8) The fact that (i) blackbody radiation, (ii) protein folding, (iii) single-molecule enzyme catalysis, (iv) wholecell RNA metabolism, and (v) T-cell receptor gene variations all fit BRE, Eq. (3), indicates that the Gibbs free
energy levels of enzymes (also called molecular machines), both in vitro and in cyto, are quantized just as the
energy levels of electrons are quantized in atoms (see Table 1 and Figure 6).
Table 1. Blackbody radiation-like equation (BRE) is obeyed by 1) blackbody radiation itself,
2) single-molecule enzymic activity of cholesterol oxidase, whole-cell RAN metabolism
measured 3) as distances between transcription rate trajectories and 4) as distances between
transcript level trajectories, 5) protein stability data, and 6) T-cell receptor CD3 domain
diveersity. The numerical values of the BRE parameters for Processes 3 and 4 are the
averages of 19 metabolic pathways with standard deviations as indicated.
y = a(Ax + B)-5/(eb/(Ax + B) - 1)
(9) The key point of Eq. (4) is that
“The rate constant of an enzyme-catalyzed reaction is determined
by the Gibbs free energy level of the ground state of the enzyme.”
10
a
b
A
B
a/b
y
x
1. Blackbody
radiation
5x10-15
4.8x10-13
1
0
1.04x10-2
Spectral
intensity
Wavelength
2. Singlemolecule
enzymology
3.5x105
2.0x102
1
0
1.75x103
Frequency of
occurrences
Waiting timea
3. Transcription
rate trajectories
(3.2±2.3)
x108
51±9.5
1.4±0.3 2.41±0.36 6.27x106
4. Distances
between RNA
trajectories
(8.8±8.9)x
108
50±11.6
2.2±1.5 3.21±1.67 1.7x107
5. Protein
stability
1.8x1010
300
6. T-cell receptor
CDR3 gene
diversity
2.3x1011
200
(10)
(11)
14
8
18
0.5
6.0x107
1.15x109
. . . . . . . . . . . . . . . . . (5)
DISCUSSION and CONCLUSIONS
(10) The Raser model of enzyme catalysis proposed in Figure 15 is built upon the quantization of Gibbs
free energy of enzymes as suggested by the fitting to BRE of the experimental data measured from
various enzyme-catalyzed processes. These findings strongly indicate that the living cell and the atom
are connected via a set of fundamental physical principles, as summarized in Table 2.
(11) One of the most interesting results listed in Table 2 is the concept of “cell orbital” in analogy to that
of “atomic orbital”.
(12) The Cell Orbital Hypothesis: Just as atomic orbitals impart structural stability to atoms, so it is
postulated here that cell orbitals impart functional stabilities (functional robustness) to the living cell.
Phenotypic similarity
classesb
Frequency of
occurrences
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)
where A’ is a constant given by the product of A and e-G‡/RT.
Table 2. The consequences of the quantization of energy levels in atoms and living cells.
Waiting time distribution, 2 mM, a=3.5*10^5, b = 200
25
. . . . . . . . . . . . . (1)
where the square brackets indicate concentrations.
Rate constant, k, of a chemical reaction, whether catalyzed or not, is given by the Arrhenius equation:
40
20
*”emission of rates” is
synonymous with “enhancement
of chemical reactions”
Figure 15. The RASER (Rate Amplification by Substrate-Enhancement of reaction Rates) model of enzyme catalysis.
(a) Unlike electrons in atoms that are all in the lowest-energy ground state before absorbing photons (Figure 6 and
14), enzymes exist in different ground states before thermal excitation.
(b) Wen an enzyme absorbs enough thermal energies through Brownian motions, it is excited to the transition state
typically lasting 10-14 to 10-12 seconds.
(c) The thermally excited enzyme undergoes a transition to the “activated state” lasting probably up to 10 -9 seconds.
(d) The activate state can be deactivated (i) spontaneously (as in “spontaneous emission” in laser, Figure 14) and (ii)
induced by substrate binding (as in “induced emission” in laser, Figure 14).
(e) During spontaneous deactivation of the activated state of an enzyme, the excess energy is released as
uncoordinated and random infrared photons, whereas during the substrate-induced deactivation, the excess energy of
the enzyme-substrate complex may be released in a coordinated manner resulting in catalysis, just as the triggering
photon-induced de-activation of electrons in atoms results in the amplification of emitted photons in laser (Figure
14).
(f) The coherent wave packets generated by substrate binding to activated enzyme may be idetified with
“conformons”, mechanical energies stored in sequence-specific sites within biopolymers [1, Chapter 8; 1a].
(g) The enzyme catalytic cycle consists of 3 steps, 1->2, 2->3, and 3->1. If the rate of steps 3->1 are determined by
(or equal to) the rate of Step 1->2, we can derive an equation for the rate constant of the enzyme reaction using the
Arrhenius equation. Rate constant, k, is defined by Eq. (1):
k = Ae-ΔG‡ /RT
(2)
B)-5)/(eb/(Ax + B)
4. Substrate-induced
emission of rates*
catalysis
heat
(1)
- 1)
3. Activated state
heat
protein folding [1] (see Panel 4)
single-molecule enzyme catalysis [1] (see Figures 5 & 6)
whole-cell RNA synthesis rates [1] (see Table 1)
whole-cell RNA levels [1] (see Figure 10)
Human T-cell receptor CDR3 gene diversification [2] (see Figure 11).
(a)
2. Transition
state
9
Phenotypic similarity
classesb
Atom
Cell
Frequency of the Δ G, the Gibbs free of
occurrence of
protein folding
ΔG
1. Particles
Electrons, protons, neutrons
Enzymes, cytoskeletons
2. Energy levels
Atomic orbitals for electrons
‘Cell orbitals’ for enzymes
Frequency of
the generation
probabilities
3. Transitions between
energy levels lead to
Absorption or emission of light
(Wavelengths)
Substrate-induced transitions of the
activated enzymes to their ground
states result in reaction rate
enhancement
(Rate constants)
4. Guaranteed stability of
Atomic structures
Cell states and functions
5. Force
Electromagnetic force mediated by photons,
and strong force mediated by gluons
Cell force mediated IDSsa and
conformonsb
Logarithm of the
generation –
probabilities of CDR3
gene sequences from
productive repertoirs
8
Figure 14. The quantum mechanical mechanisms
underlying laser, Light-Amplification by
Stimulated Emission of Radiation.
(a) The input of “pumping” photons, hν1, causes
the electrons of the atoms constituting the laser
medium (e.g., ruby crystal) to undergo a
transition from the ground-state energy level to
the excited-state energy level (from 1 to 2).
(b) The excited state lasts for only 10-12 seconds
or less, loses some of its energy as heat and
undergoes a transition to a lower energy level
called “metastable” state (from 2 to 3).
(c) When there are enough number of electrons
in the excited state (“population inversion”), the
arrival of triggering photons, hν2, induces the
de-excitation of electrons from the metastable
excited state to the ground state (from 3 to 1),
accompanied by the emission of photons
identical to the triggering photons, hν2, but
larger in number than the original triggering
photons.
(d) The emitted photons are “coherent” in in
that they are identical with respect to (i)
amplitude, (ii) frequency, and (ii) phase.
aIntracellular
Dissipative Structures , e.g., membrane potential, RNA gradients; Ca++ gradients
bMechanical energy stored in site-specific regions within biopolymers [1a].
11
ACKNOWLEDGEMENT
I gratefully acknowledge the contributions made by the past and current students in my Theoretical and
Computational Cell Biology Lab at the Ernest Mario School of Pharmacy, Rutgers University, toward
developing the RASER model of enzyme catalysis. I am particularly grateful to Mr. Kenneth So who
wrote the computer program for finding the best-fit blackbody radiation-like equation (BRE) based on the
genome-wide RNA kinetic data.
REFERENCES
[1] Ji, S. (2012). Molecular Theory of the Living Cell: Concepts, Molecular Mechanisms, and Biomedical Applications, Springer, New York. See
Sections 11 and 12.
[1a]. Ji, S. (2000). Free energy and Information Contents of Conformons in proteins and DNA, BioSystems 54, 107-130.
[2] Murugan, A. et al. (2012). Statistical inference of the generation probability of T-cell receptors from sequence repertoires. ArXiv:1208.3925v1
[q-bio.QM] 20 Aug 2012.
[3] Watson, S. J. and Akil, U. (1999). Gene Chips and Arrays Revealed: A Primer on their Power and Their uses, Biol. Psychiatry 45, 533-43.
[4] Lu, H. P., Xun, L. and Xie, X. S. (1998). Single-Molecule Enzymatic Dynamics, Science 282, 1877-1882.
[5] Garcia-Martinez, J., Aranda, A. and Perez-Ortin, J. E. (2004). Genomic Run-On Evaluates Transcription Rates for all Yeast Genes and
Identifies Gene Regulatory Mechanisms, Mol Cell 15, 303-313.
[6] Ji, S. and So, K. (2009). The universal law of thermal transitions applicable to blackbody radiation, single-molecule enzymology and wholecell metabolism, Abstract B1, The 102nd Statistical Mechanics Conference, Rutgers University, Piscataway, N.J., December 13-15.
[6a] Watson, S. J. and Akil, U. (1999). Gene Chips and Arrays Revealed: A Primer on their Power and Their uses, Biol. Psychiatry 45, 533-43.
[7] Ji, S. (2008). Modeling the single-molecule enzyme kinetics of cholesterol oxidase based on Planck's radiation formula and the principle of
enthalpy-entropy compensation, in Short Talk Abstracts, The 100th Statistical Mechanics Conference, December 13-16, Rutgers University,
Piscataway, N.J.
[8] Frauenfelder, H., McMahon, B. H., Austin, R. H., Chu, K. and Groves, J. T. (2001). The role of structure, energy landscape, dynamics, and
allostery in the enzymatic function of myoglobin, Proc. Nat. Acad. Sci. (U.S.) 98(5), 2370-74.