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BMED 3510
Multi-Scale Models
The Heart as Example
Book Chapter 12
What Does “Multi-Scale” Mean?
Length, Time, and Functional Scales
years
mosquitos
days
human
cells
hours
minutes
seconds
bacteria,
parasites
human
organs
ecosystems
Heart
large
molecules,
viruses
Calcium
small
milliseconds molecules
1nm
1mm
1mm 1cm
1m
1km
Issues
Different “players” at different levels
Number of players large at each level
Combinatorial explosion if all lower levels are retained in detail
E.g., number of molecules involved in heart function is enormous.
Not feasible to describe the function, aging, and diseases of the heart by
accounting for all molecular processes all the time.
Most relevant in physiology: Physiome Project (http://physiomeproject.org/)
Current “Solutions”
Focus on one or two levels
Use only one (most appropriate) time scale
3
Separation of Time Scales
Implementation
(Theoretically) start with large system with many variables
All very slow processes are considered constant (dX / dt = 0)
All very fast processes are assumed to be in steady state (dX / dt = 0)
Retain only processes at just the right time scale of interest
4
Different Models of the Same Subject
Examples
Light:
Light as wave or light as “corpuscles” (little particles)
Heart:
Pump
Big muscle
Oscillator
Chemical system (Ion flow)
Electrical system
Very many examples, not just of organs, but:
Cancer, infectious diseases, inflammation, ecosystems, …
5
Stats of the Heart
Amazing facts
Was (is?) thought to be the location of
love and emotions, no “broken brain”
Size of a fist; weighs about half a pound
Moves blood through a network of about
60,000 miles of arteries, veins and
capillaries
Pumps over 7,000 liters of blood per day
Beats roughly 100,000 times every day
Beats 2 to 3 billion times in a normal life
time
Miss a few beats in a row: game over!
6
The Heart as a Peristaltic Pump
http://www.dovercorporation.com/Image%20Library/DoverCorpImages/articles-engineered-systems/Abaque-Pump7
components.jpg?code=3cf4e99f-b1e7-46d3-b28e-d9fd9931a89d, http://www.thevintnervault.com/category/435/Peristaltic-Pumps.html
The Heart as a Very Sophisticated Pump
Actually: Two pumps that drive different portions of the circulation of blood
(to the lungs; to the rest of the body)
In contrast to almost all engineered pumps:
Sensitive and adaptable to momentary needs
Constantly adjusts influx – efflux in response to demands
Adjustments are “autonomous”
Increasing the venous blood input to the ventricle stretches the
ventricular wall, enhances contractility, and elevates the diastolic
pressure and volume of the ventricle, which in turn leads to an
increased stroke volume
Very complicated fluid dynamics
Long-term adaptation possible
8
The Heart as a Muscle Systems
9
www.helicalheart.com/index_files/h3.gif; www.knowyourbody.net/wp-content/uploads/2012/06/Papillary-muscle-Picture.gif
The Heart as a Muscle Systems
10
The Heart as a Muscle Systems
Imagine what it would take to develop
a dynamic model describing the heart
on the basis of contraction and
relaxation of myofibrils!
11
The Heart as an Chemical Systems
Repeated contraction and relaxation of heart cells associated with:
Periodic movement of calcium ions between three locations:
extracellular fluid
cytosol, and intracellular
sarcoplasmic reticulum (SR)
12
The Heart as an Chemical Systems
13
The Heart as an Chemical Systems
1. Electrical signal: influx of
sodium; small amount of calcium
flows from the extracellular space
into the cytosol
2. Calcium triggers a mechanism
called calcium-induced calcium
release: results in the flow of
large amounts of calcium from
the SR into cytosol
3. Calcium binds to troponin;
leads to sliding action of the
myfibrils actin and myosin
4. Myocyte contracts
5. Calcium unbinds and is
pumped back into the SR; cell
relaxes.
14
The Heart as an Electrical Systems
Normal heart beat
Electrical signals that start
from the Sinoatrial node (SA
node)
Move to the Atrioventricular
Node (AV node).
Moves down the Bundle of
His in the septum and
Purkinje fibers.
Electrical signal causes
contraction first of the
atrium and then the
ventricles.
15
The Heart as an Electrical Systems
Normal heart beat
Electrical signals that start
from the Sinoatrial node (SA
node)
Move to the Atrioventricular
Node (AV node).
Moves down the Bundle of
His in the septum and
Purkinje fibers.
Electrical signal causes
contraction first of the
atrium and then the
ventricles.
16
Rhythm of the Electrical System
PR
Segment
QRS
Complex
ST
Segment
PR
Interval
1 mV
QT
Interval
0
0.2
0.4
0.6
Time (sec)
0.8
17
http://www.austincc.edu/apreview/PhysText/Cardiac.html; WikiCommons File ECG-PQRST+popis.svg
The Electrical System Oscillates
Contraction and relaxation switch off regularly: Oscillations
Why do we see different oscillatory patterns in an EKG?
18
Myocardial Infarction
Problems in electro-chemical activity lead to fibrillation
Many possible reasons, including genetics
19
Myocardial Infarction
20
http://thevirtualheart.org/FentonCherry/Newecg_n_vt_vf.s2.gif
Heart Models: Pure Math
(x  y  1)  x y  0
2
2
3
2 3
(x 2  (9 / 4)y 2  z 2  1)3  x 2 z 3  (9 /80)y 2 z 3  0
3  x , y , z  3
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Modeling a Peristaltic Pump
Instantaneous volumetric flow rate:
Pressure drop:
22
Linear Oscillator Models
23
Nonlinear
Oscillator
Models
Limit Cycle
Model of van
der Pol (1928)
v  k (1  v 2 )v  v  0
Write as two
first-order
ODEs:
w  k (1  v 2 ) w  v
vw
24
Nonlinear Oscillator Models
Flexible Alternative for Limit Cycles: S-system models (Yin and Voit; 2008)
X 1  2.5  ( X 26  X 13 X 23.2 )
X 2  1.001  2.5  (1  X 15 X 23 )
X 1  1.005  ( X 28  X 17 X 25 )
X 2  X 16 X 25  X 11 X 24
25
Toward Fibrillation
S-system model B, “poked” with a sine function (spurious signal with frequency A)
A=0
A = 10
26
Toward Fibrillation
A = 0.5
A = 0.42
27
Toward Fibrillation
A = 0.4
A = 0.394
28
Calcium Flow
29
Calcium Flow Model
rEC
y = [Ca2+]cyt
ctSC f(y)
pCS
rSC
x = [Ca2+]SR
x  pCS y  rSC ( x  y)  ct SC
y  rEC  rSC ( x  y)  ct SC
pCE
yh
( x  y)
h
h
K y
yh
( x  y)  pCE y  pCS y
h
h
K y
30
Models of Action Potentials
Hodgkin and Huxley (1952) showed
that an action potential can be
modeled as an electrical circuit
Important difference to
electrical circuits
The lipid bilayer
~ capacitance (Cm)
Voltage-gated and leak ion
channels
~ nonlinear (gn) and linear (gL)
conductances
Electrochemical gradients
driving the flow of ions
~ batteries (E)
Ion pumps and exchangers
~ current sources (Ip).
31
File:Hodgkin-Huxley.svg
Models of Action Potentials
dn
  n (v)(1  n)   n (v)n
dt
dm
  m (v)(1  m)   m (v)m
dt
dh
  h (v)(1  h)   h (v)h
dt
Voltage (mV)
100
45
-10
0
10
20
time [msec]
Voltage (mV)
100
+ calcium oscillation model
45
-10
0
40
time (msec)
80
32
Heart Disease
Numerous causes
Aging, stress, hereditary factors
Always molecular events involved
Need to span the gap from genes and ions to malfunctioning physiology
33
Multi-Scale Heart Disease
Raymond Winslow’s group
(Johns Hopkins) connected:
specific gene mutation
to malfunction of dyad
to malfunction of calcium release
to formation to spiral waves
to fibrillation
34
Summary of Heart Chapter
Multi-scale models pose an unsolved problem
Current approaches:
Isolated models at different levels
Bridging two levels
Mesoscopic models as starting points
Template-and-anchor models
(coarse high-level template models;
fine-grained anchor models of selected details)
Hybrid agent-based models with dynamic (ODE) submodels
35
Frontiers of Systems Biology
Modeling Needs
Data pipelines
Automation of translation from biology to models
Multi-scale modeling
More effective hybrid modeling
(e.g., ODE + ABM for spatial features and stochasticity)
Improved guidelines for choosing optimal mathematical formats
Parameter estimation
Theory of biology (why?)
36
Frontiers of Systems Biology
The Brain
Several hundred cell types in human brain
100 billion neuronal components
100s of trillions of interconnections between neurons
Multi-scale combination of electrical and chemical
activities within complex anatomy
Emerging features such as cognition, learning,
memory, consciousness
Start with Caenorhabditis elegans with 959
somatic cells and 302 neurons?
37
Frontiers of Systems Biology
The Immune System
Actually two systems: innate and adaptive (acquired) immune systems
Complex generation of cells responding to new agents like pathogens
Huge variability in antibodies
Confusing system of cytokines
Immunological memory
Inflammatory and autoimmune diseases
Immunodeficiency, cancer
38
Frontiers of Systems Biology
The Immune System
39
Frontiers of Systems Biology
Metapopulations
Usually microbial
Examples: Gut, skin, oral cavity; soil, lakes, moist environments
Often thousands of species (definition of species in question)
Complicated biofilms; quorum sensing
Unclear how important rare species are
Species compete for resources and depend on each other
Most species cannot be cultured in the lab, so how can we study them?
40
Frontiers of Systems Biology
Whole cell models
Account for all types of molecules
Markus Covert’s group (Stanford):
“A whole-cell computational model
predicts phenotype from genotype,”
Cell 150, 2012
Very simple cell (Mycoplasma genitalium)
Fifteen person-years to construct
this model
Our Hope in BMED 3510
…is that:
You have learned how to approach complexity in biology with
computational means
You recognize that all biological components are parts of systems
You realize that our intuition is not sufficient to analyze complex systems,
even if they are fairly small
You have started to see the biological world with different eyes
42
Wrap-Up
Please:
Participate in CIOS
Be constructive
Send us emails with
additional suggestions
Send problems with the book
Good Luck with the Final!