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Group: Deep Thought
Transfer of Heat in Biological Systems
• Tissue
– Skin
• 3 layers:
– A thin outer Epidermis
– A thicker layer of Dermis
– A thick subcutaneous fatty tissue (Hypodermis)
• First-, Second-, and Third-Degree Burns
• How is heat transferred in the system?
• Temperature Field
– Three-dimensional
– Non-uniform
• So…
– A mathematical formulation of burns is
difficult to obtain
• But….
– Difficult, but not impossible
T
'''
c
 kt  b  b cb T  Tb   qm
t
•  is density
• c is heat capacity
• k is the conductivity of tissue
• "b" is an index for the physical properties of blood
• b is the normalized blood perfusion of tissue
• qm''' is the heat from metabolism
• T is the body core or deep tissue temperature
• Tb is the temperature of the skin at the burn site
Cellular Level
Free Energy Distributions
Thermal Protein Denaturation
Burn Trauma at the Cellular Level
• Which cellular structures are most
vulnerable?
• Which are most critical in cell viability?
• Kinetics of damage ought to depend on
multiple factors in the chemical
environment
• Moritz and Henriques (1947):
Time – Temperature relationship for
scald burning of forearm skin was
Arrhenius
Arrhenius Process
• Observed in thermally activated reactions
• Collision Theory:
Molecules react if they collide with kinetic
energy that exceeds EA
• Explains temperature dependence of
reaction rates
• Boltzmann: probability of reactive
collisions
 E A / RT
k  k0 e
Application of Arrhenius
• How is Thermal Injury explained by Arrhenius?
• Two Hypotheses:
1.
Statistical Mechanics.
“Central Limit Theorem”
2.
Specific Denaturation.
Statistical Thermodynamics
Free Energy Distribution in Proteins
Energetics of Denaturation
• Thermodynamics forces drive changes
in protein conformations
• Central Limit Theorem: infinite number
of different processes behave like single
Arrhenius process
• Gibbs Free Energy (G): max energy
available for work
Free Energy Distribution in Proteins
• Douglas Poland
• Protein in aqueous solutions
– Fluctuations in conformation and molecular
vibrations  broad distribution of enthalpy
states
– Approximate distribution function using
maximum – entropy method from moments
• Gm becomes the central function:
describes thermal behavior of a protein in
the enthalpy neighborhood of the
denaturation maximum.
G( H ) Gm  H  1 1 

    
kT
kTm  k  T Tm 
•
At normal physiological temperatures,
proteins are in a folded, threedimensional conformation.
•
When a burn injury occurs, tissue is
heated to well above physiological pH,
and components of a protein’s 3-D
structures are damaged, resulting in an
unfolding and eventual denaturing of
the protein.
•
Denaturation is irreversible.
•
A simple example of protein
denaturation is the cooking of an egg
white.
(F. Despa, 2005)
•
•
•
•
•
•
The process of protein denaturation and aggregation can be modeled as a
statistical process
Rate of protein unfolding similar to the Arrhenius equation
As the temperature increases, a protein is unable to remain in its normal, folded
conformation and begins to transition into its unfolded state.
At the melting temperature of the protein, it can unfold and refold at the same
rate.
The protein can also reach its denatured state by transitioning irreversibly from its
unfolded state
If the rate of conversion from the unfolded state to the denatured state is faster
than the transition from the unfolded state to the refolded state, then the rate of
denaturation becomes independent of the rate of irreversible unfolding and
refolding.
Rate of unfolding and refolding:
• K≡
ku
e
kf
(
H
T
(1 ))
RT
Tm
Percent denaturation equation:
K (T )
fd 
1  K (T )(F. Despa, 2005)
•
The kinetics of protein denaturation is affected by so many different factors: density,
solvent, bond strengths, interactions with surrounding molecules.
•
Since every protein is different, then clearly each protein should have a different rate of
denaturation.
•
How, then, is it possible that the rate of protein denaturation follows a single, Arrheniuslike equation?
•
2 hypotheses:
1.
Cell denaturation depends on an infinite number of different processes that, when
combine, behaves like a single Arrhenius process.
2.
Lethal burn injury is dominated by only a few molecular processes i.e., there are one
or two key structures that are critical to cell survival in a burn injury.
•
The relative stability of a variety of different types of tissues was investigated by F. Despa et
al. to determine if cell denaturation is dependent on only a few molecular processes.
(F. Despa, 2005)
Table.1:Percent
denaturation of proteins
and cellular components
after 20 seconds exposure
to varying temperatures
(40-66ºC)
Table 2: Percent
denaturation of
proteins and
cellular
components at
80 ºC as a
function of time.
(F. Despa, 2005)
•
As can be seen from the
graphs, most of proteins in
the study denature at around
60 ºC.
•
The lipid bilayer and the
membrane bound ATPases,
the Na+/K+ pump (NKP) and
Ca2+ pump (PMCP), are the
first to denature.
•
As a result of these findings,
this study suggests that
alteration of the plasma
membrane and its
components as a result of
high temperatures “is likely to
be the most significant cause
of tissue necrosis.”
(F. Despa, 2005)
http://academic.brooklyn.cuny.edu/biology/bio4fv/page/cotrans.htm
• Heating therapies that intentionally
incite protein denaturation are being
used in a variety of medical fields
• Most of these therapies are refined by
trial and error
• Developing theoretical models
– Reduces need for extensive clinical trials
– Makes therapies more effective
• Determine rate of denaturation
– As a function of temperature
– As a function of mechanical load
• Determine the values of thermophysical
properties
– Specific Heat
– Thermal Conductivity
– Thermal Diffusivity
• Target of many heating therapies
• Triple-helix structure
• Moderate heating
– Induces reversible local unfolding
– Breaking of a few hydrogen bonds
– Regains shape upon cooling
• Severe heating
– Time-dependent irreversible changes
– Breaking of many hydrogen bonds
– Random, coiled structure
• Shrinks upon heating
• Quickly heat collagen to a specific
temperature
• Measure shrinkage over time
isothermally
• Equation:
 K (T )
w  e
• ξ is the shrinkage, K(T) is the specific
reaction rate, τ is the time
“Denaturation of Collagen Via Heating: An Irreversible Rate Process” Wright
•
•
•
Actually developing a working
mathematical model is beyond the
scope of this project
• Requires fairly extensive empirical
validation and assigning meaningful
values to constants
• Our proposal for animal testing fell
through, so we were unable to
gather our own data on this front
Solve:
T
c
 kt  b  b cb T  Tb   qm'''
t
For temperatures, then plug that result
into the Arrhenius
 Ea
An initial heat distribution resulting from
point exposure to a high temperature object
d
 Ae RT
dt
•
•
Both equations must be integrated
numerically.
As the reaction proceeds heat is
absorbed and released by the proteins
folding and unfolding.
Heat diffusion as experienced by a
particular skin surface model.