The physics of obesity. Carson Chow Laboratory of Biological Modeling, NIDDK, NIH
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1 The physics of obesity Carson Chow Laboratory of Biological Modeling, NIDDK, NIH
2 US obesity epidemic 82 Average Body Weight (kg) Year Data from National Health and Nutrition Examination Survey (NHANES)
3 US obesity epidemic BMI NHANES data
4 US obesity epidemic BMI = weight / height 2 NHANES data
5 US obesity epidemic obese BMI = weight / height 2 NHANES data
6 US obesity epidemic obese BMI = weight / height 2 NHANES data
7 US obesity epidemic obese BMI = weight / height 2 NHANES data
8 US obesity epidemic obese BMI = weight / height 2 NHANES data
9 Conservation of energy
10 Conservation of energy Food Intake
11 Conservation of energy Food Intake Energy expenditure
12 Conservation of energy Food Intake Energy Energy storage expenditure
13 Conservation of energy Food Intake Energy Energy storage expenditure Storage = Intake - Expenditure
14 Energy flux Rate of storage = intake rate - expenditure rate d(ρ M M) = I E M = body mass Energy density ρm converts energy to mass
15 Energy density Fat Water 37.7 kj/g Carbs (glycogen) Bone 16.8 kj/g Protein Minerals 16.8 kj/g
16 Multiple fuel sources IF IC IP F G P E Fat Carbs/ glycogen Protein { Macronutrients
17 Macronutrient flux d(ρ M M) = I E
18 Macronutrient flux ρ F df dp + ρ P + ρ G dg = I E
19 Macronutrient flux ρ F df + dp dg ρ P ρ G = E + I F + I C + I P
20 Macronutrient flux ρ F df ρ G dg = I F + I C + I P E ρ P dp
21 Macronutrient flux ρ F df = I F ρ G dg = E I C ρ P dp = I P
22 Macronutrient flux ρ F df = I F f F E ρ G dg = I C f C E ρ P dp = I P (1 f F f C )E
23 Macronutrient flux ρ F df = I F f F E ρ G dg = I C f C E ρ P dp = I P (1 f F f C )E ff = fraction of fat utilized fc = fraction of carbs utilized
24 Macronutrient flux ρ F df = I F f F E ρ G dg = I C f C E ρ P dp = I P (1 f F f C )E ff = fraction of fat utilized fc = fraction of carbs utilized
25 Reduction to 2D ρ F df = I F f F E ρ G dg = I C f C E ρ P dp = I P (1 f F f C )E ff = fraction of fat utilized fc = fraction of carbs utilized
26 Reduction to 2D ρ F df = I F f F E ρ G dg = I C f C E ρ P dp = I P (1 f F f C )E Glycogen supply small, ~ fixed on long time scales
27 Reduction to 2D ρ F df = I F f F E ρ G dg = I C f C E =0 ρ P dp = I P (1 f F f C )E Glycogen supply small, ~ fixed on long time scales
28 Reduction to 2D ρ F df = I F f F E I C E f C = ρ P dp = I P (1 f F f C )E Glycogen supply small, ~ fixed on long time scales
29 Reduction to 2D ρ F df = I F f F E ρ P dp = I P (1 f F I C )E E Glycogen supply small, ~ fixed on long time scales
30 Reduction to 2D ρ F df = I F f F E ρ P dp = I P (1 f F )E +I C Glycogen supply small, ~ fixed on long time scales
31 Reduction to 2D ρ F df = I F f F E ρ P dp = I P +I C (1 f F )E Glycogen supply small, ~ fixed on long time scales
32 Reduction to 2D ρ F df = I F f F E ρ P dp = I P +I C (1 f F )E Divide mass into lean and fat M = L + F
33 Reduction to 2D ρ F df = I F f F E ρ P dp = I P +I C (1 f F )E Divide mass into lean and fat M = L + F Change in L due to change dp 1 dl in P and water = 1+h P
34 Reduction to 2D ρ F df = I F f F E ρ P dl = I P +I C (1 f F )E 1+h P Divide mass into lean and fat M = L + F
35 Reduction to 2D ρ F df = I F f F E ρ P dl = I P +I C (1 f F )E 1+h P Divide mass into lean and fat M = L + F Lean intake = carbs + protein I P + I C = I L
36 Reduction to 2D ρ F df = I F f F E ρ P dl +I C (1 f F )E 1+h P = I P I L Divide mass into lean and fat M = L + F
37 Body composition model df ρ F dl ρ L = I F fe = I L (1 f)e
38 Body composition model df ρ F dl ρ L = I F fe = I L (1 f)e ρ L = ρ P /(1 + h P ) hp protein hydration coefficient
39 Body composition model df ρ F dl ρ L = I F fe = I L (1 f)e ρ L = ρ P /(1 + h P ) hp protein hydration coefficient f is fraction of energy use that is fat
40 Body composition model df ρ F dl ρ L = I F fe = I L (1 f)e ρ L = ρ P /(1 + h P ) hp protein hydration coefficient f is fraction of energy use that is fat E and f are functions of F and L
41 Dynamical systems Infer global dynamics from local information
42 Dynamical systems dx = x(1 x)
43 Dynamical systems dx = x(1 x) Fixed = 0 points
44 Dynamical systems dx = x(1 x) Fixed = 0 points x = 0, 1
45 Dynamical systems dx = x(1 x) Fixed = 0 points x = 0, 1 Represent geometrically in phase plane 0 1 x
46 Dynamical systems dx = x(1 x) Fixed = 0 points x = 0, 1 Represent geometrically in phase plane 0 dx 1 > 0 x
47 Dynamical systems dx = x(1 x) Fixed = 0 points x = 0, 1 Represent geometrically in phase plane 0 dx 1 > 0 dx < 0 x
48 Dynamical systems dx = x(1 x) Fixed = 0 points x = 0, 1 Represent geometrically in phase plane 0 dx 1 > 0 vector field dx < 0 x
49 Fixed points ρ F df = I F fe ρ L dl = I L (1 f)e
50 Fixed points field{ vector ρ F df = I F fe ρ L dl = I L (1 f)e
51 Fixed points Nullclines field{ vector ρ F df = I F fe ρ L dl = I L (1 f)e = 0 = 0 L - Nullcline F - Nullcline
52 Fixed points Nullclines field{ vector ρ F df = I F fe ρ L dl = I L (1 f)e = 0 = 0 L - Nullcline F - Nullcline E(F, L) =I
53 Fixed points Nullclines field{ vector ρ F df = I F fe ρ L dl = I L (1 f)e = 0 = 0 L - Nullcline F - Nullcline E(F, L) =I energy balance
54 Fixed points Nullclines field{ vector ρ F df = I F fe ρ L dl = I L (1 f)e = 0 = 0 L - Nullcline F - Nullcline E(F, L) =I energy balance f(f, L) = I F I
55 Fixed points Nullclines field{ vector ρ F df = I F fe ρ L dl = I L (1 f)e = 0 = 0 L - Nullcline F - Nullcline E(F, L) =I f(f, L) = I F I energy balance fat balance
56 Phase plane I L (1 f)e(f, L) = 0 L - nullcline F I F fe(f, L) = 0 F - nullcline L
57 Phase plane I L (1 f)e(f, L) = 0 L - nullcline F I F fe(f, L) = 0 F - nullcline L
58 Phase plane I L (1 f)e(f, L) = 0 L - nullcline F df I F fe(f, L) = 0 F - nullcline L
59 Phase plane I L (1 f)e(f, L) = 0 L - nullcline F df I F fe(f, L) = 0 F - nullcline L
60 Phase plane I L (1 f)e(f, L) = 0 L - nullcline F df dl I F fe(f, L) = 0 F - nullcline L
61 Phase plane I L (1 f)e(f, L) = 0 L - nullcline F df dl I F fe(f, L) = 0 F - nullcline L
62 Phase plane I L (1 f)e(f, L) = 0 L - nullcline F df dl I F fe(f, L) = 0 F - nullcline L
63 Phase plane I L (1 f)e(f, L) = 0 L - nullcline F df dl I F fe(f, L) = 0 F - nullcline L
64 Phase plane I L (1 f)e(f, L) = 0 L - nullcline F df dl I F fe(f, L) = 0 F - nullcline Fate of all initial conditions is known L
65 Possible phase plane dynamics a) c) F F Fixed point L Line attractor L b) d) F F L L Multiple fixed points Limit cycle Chow and Hall, PLoS Comp Bio,4: e , 2008
66 Indirect calorimetry Carbohydrates: C6H12O6 + 6 O2 6 CO2 + 6 H2O Fat: C16H32O O2 16 CO H2O Protein: C72H112N2O22S + 77 O2 63 CO H2O + SO3 + 9 CO(NH2)2 Flux of CO2 and O2 E
67 Energy expenditure rate E E= + Basal metabolic rate (BMR) Physical activity
68 Energy expenditure rate E E= + Basal metabolic rate (BMR) E ~ 10 MJ/day Physical activity
69 Energy expenditure rate E E= + Basal metabolic rate (BMR) E ~ 10 MJ/day ~115 W Physical activity
70 Energy expenditure rate E E= + Basal metabolic rate (BMR) Physical activity E ~ 10 MJ/day ~115 W ~ 3 KWH/day
71 Basal metabolic rate Nielson, 2000
72 Basal metabolic rate Nielson, 2000 e.g. BMR (MJ/day) = 0.9 L (kg) F (kg) +1.1
73 Physical activity Energy due to PA Mass E P A = am = a(l + F ) a ranges from 0 to 0.1 MJ/kg/day
74 Physical activity Energy due to PA Mass E P A = am = a(l + F ) a ranges from 0 to 0.1 MJ/kg/day E is linear in F and L
75 E(F, L) =bf + cl + d = I f(f, L) = I F I F L
76 E(F, L) =bf + cl + d = I f(f, L) = I F I F L Single fixed point is generic
77 E(F, L) =bf + cl + d = I F F + ql 3 + rl 2 + sl + t = I F E F L Multi-stability or limit cycle requires fine tuning
78 E(F, L) =bf + cl + d = I f = I F E + ψ(i E) F L Line attractor requires special form
79 E(F, L) =bf + cl + d = I f = I F E + ψ(i E) F L Line attractor requires special form
80 The problem with f In energy balance, f reflects diet
81 The problem with f In energy balance, f reflects diet f(f, L) = I F I
82 The problem with f In energy balance, f reflects diet f(f, L) = I F I Must invert in dynamic situation df ρ F dl ρ L = I F fe = I L (1 f)e
83 Body composition L F Forbes, 2000
84 Body composition L F Forbes, 2000
85 Apply Forbes law to model df dl = F 10.4 ρ F df = (I F fe) ρ L dl = (I L (1 f)e)
86 Apply Forbes law to model df dl = F 10.4 df = (I F fe) ρ F ρ L dl = (I L (1 f)e)
87 Apply Forbes law to model df dl = F 10.4 df = (I F fe) ρ F dl = (I L (1 f)e) ρ L
88 Apply Forbes law to model df dl = F 10.4 df dl = (I F fe) ρ L (I L (1 f)e) ρ F
89 Apply Forbes law to model (I F fe) ρ L = F (I L (1 f)e) ρ F 10.4
90 Apply Forbes law to model (I F fe) ρ L = F (I L (1 f)e) ρ F 10.4 f = I F (1 p)(i E) E p = ρ F ρl F 10.4
91 Apply Forbes law to model (I F fe) ρ L = F (I L (1 f)e) ρ F 10.4 f = I F (1 p)(i E) E p = ρ F ρl F 10.4
92 Matches data 2000 f E Energy Rate (kcal/d) (1-f)E IL IF Time (days) Hall, Bain, and Chow, Int J. Obesity, (2007)
93 Weight and fat loss 0 Mass Change (kg) Time (days) Hall, Bain, and Chow, Int J. Obesity, (2007)
94 Energy partition model ρ F df = ρ L dl = I F f E I L (1 f)e f = I F (1 p)(i E) E
95 Energy partition model ρ F df = ρ L dl = I F I F (1 p)(i E) E I L (1 f)e E
96 Energy partition model ρ F df = ρ L dl = I F I F + (1 p)(i E) I L (1 f)e
97 Energy partition model ρ F df = (1 p)(i E) ρ L dl = I L (1 f)e
98 Energy partition model ρ F df = (1 p)(i E) ρ L dl = p(i E)
99 Energy partition model ρ F df = (1 p)(i E) ρ L dl = p(i E) Steady state is line attractor E(F, L) = I
100 Energy partition model ρ F df = (1 p)(i E) ρ L dl = p(i E) Steady state is line attractor E(F, L) = I Almost all previous models use energy partition
101 Consequences of line attractor E(F,L)=I L F
102 Consequences of line attractor E(F,L)=I L possible histories F
103 Consequences of line attractor E(F,L)=I L possible histories constant weight F + L = M F
104 Consequences of line attractor E(F,L)=I possible histories L Change lifestyle constant weight F + L = M F
105 Consequences of line attractor E(F,L)=I possible histories L Change lifestyle constant weight F + L = M F
106 Effect of perturbations Perturbed a) b) Perturbed F Original F Original L L fixed point line attractor
107 Numerical example df ρ F dl ρ L = I F fe = I L (1 f)e ρf = 37.7 MJ/kg ρl = 7.6 MJ/kg E (MJ/Day) =0.14 L (kg)+0.05 F (kg) +1.55
108 Numerical example df ρ F dl ρ L = I F fe = I L (1 f)e ρf = 37.7 MJ/kg ρl = 7.6 MJ/kg E (MJ/Day) =0.14 L (kg)+0.05 F (kg) f = I F E (1 p)i E E ψ E
109 Numerical example df ρ F dl ρ L = (1 p)(i E)+ψ = p(i E) ψ ρf = 37.7 MJ/kg ρl = 7.6 MJ/kg E (MJ/Day) =0.14 L (kg)+0.05 F (kg) +1.55
110 Numerical example df ρ F dl ρ L = (1 p)(i E)+ψ = p(i E) ψ ρf = 37.7 MJ/kg ρl = 7.6 MJ/kg E (MJ/Day) =0.14 L (kg)+0.05 F (kg) Forbes: p = 2/(2 + F )
111 Numerical example df ρ F dl ρ L = (1 p)(i E)+ψ = p(i E) ψ ρf = 37.7 MJ/kg ρl = 7.6 MJ/kg E (MJ/Day) =0.14 L (kg)+0.05 F (kg) Forbes: p = 2/(2 + F ) ψ = { 0.05(F 0.4 exp(l/10.4))/f 0
112 Numerical example df ρ F dl ρ L = (1 p)(i E)+ψ = p(i E) ψ ρf = 37.7 MJ/kg ρl = 7.6 MJ/kg E (MJ/Day) =0.14 L (kg)+0.05 F (kg) Forbes: p = 2/(2 + F ) fixed point ψ = { 0.05(F 0.4 exp(l/10.4))/f 0 invariant manifold
113 a) c) Mass (kg) Mass (kg) Days Days b) d) F (kg) F (kg) L (kg) L (kg)
114 Mean US body weight 82 Average Body Weight (kg) Year Data from National Health and Nutrition Examination Survey (NHANES)
115 Mean US body weight Average Body Weight (kg) ρ F df = (1 p)(i E) dl ρ L = p(i E) Year Data from National Health and Nutrition Examination Survey (NHANES)
116 Mean US body weight 82 ρ L dl df ρ F + = I E Average Body Weight (kg) Year Data from National Health and Nutrition Examination Survey (NHANES)
117 Mean US body weight 82 E + ρ L dl + ρ F df = I Average Body Weight (kg) Year Data from National Health and Nutrition Examination Survey (NHANES)
118 Mean US body weight 82 E + ρ L dl + ρ F df = I Average Body Weight (kg) Year Data from National Health and Nutrition Examination Survey (NHANES)
119 Mean US body weight 82 E + ρ L dl + ρ F df = I Average Body Weight (kg) Year Data from National Health and Nutrition Examination Survey (NHANES)
120 U.S. Food Supply FAO Food Supply Food Intake (model) Hall, Guo, Dore, Chow. PLoS One (2009)
121 U.S. Food Supply FAO Food Supply USDA Food Availability Hall, Guo, Dore, Chow. PLoS One (2009)
122 U.S. Food Supply FAO Food Supply USDA Food Availability Predicted Intake Hall, Guo, Dore, Chow. PLoS One (2009)
123 U.S. Food Supply FAO Food Supply USDA Food Availability Predicted Intake Hall, Guo, Dore, Chow. PLoS One (2009)
124 U.S. Food Supply FAO Food Supply ~1400 kcal USDA Food Availability Predicted Intake Hall, Guo, Dore, Chow. PLoS One (2009)
125 U.S. Food Waste Food Waste (model) Food Waste (USDA) Hall, Guo, Dore, Chow. PLoS One (2009)
126 U.S. Food Waste Food Waste (model) Solid Food Waste (EPA) Hall, Guo, Dore, Chow. PLoS One (2009)
127 One dimensional models Energy partition models are effectively 1D df ρ F dl ρ L = (1 p)(i E) = p(i E) L m = ρ F 1 p ρ L p F
128 One dimensional models Energy partition models are effectively 1D ρ L dl df ρ F + = I E m = ρ F 1 p ρ L p L F
129 One dimensional models Energy partition models are effectively 1D ρ L dl df ρ F + = I E m = ρ F 1 p ρ L p L L mf + b F
130 One dimensional models Energy partition models are effectively 1D ρ L dl df ρ F + = I E m = ρ F 1 p ρ L p L L mf + b F = M L F
131 One dimensional models Energy partition models are effectively 1D ρ L dl df ρ F + = I E m = ρ F 1 p ρ L p L L mf + b F = M L F = M b 1 + m F
132 One dimensional models Energy partition models are effectively 1D ρ L dl df ρ F + = I E m = ρ F 1 p ρ L p L L mf + b L = mm + b 1 + m F = M L F = M b 1 + m F
133 One dimensional models ρ M dm = I ɛm b ρ M = ρ F ρ L ρ L +(ρ F ρ L )p
134 One dimensional models ρ M dm = I ɛm b ρ M = ρ F ρ L ρ L +(ρ F ρ L )p Leaky integrator
135 One dimensional models ρ M dm = I ɛm b ρ M = ρ F ρ L ρ L +(ρ F ρ L )p Leaky integrator C dv = I 1 R V
136 Steady state ρ M dm = I ɛm b = 0 M =(I b)/ɛ M 1 ɛ I
137 Steady state ρ M dm = I ɛm b = 0 M =(I b)/ɛ M 1 ɛ I ε decreases with weight and increases with activity
138 Steady state ρ M dm = I ɛm b = 0 M =(I b)/ɛ M 1 ɛ I ε decreases with weight and increases with activity ε ~ 0.1 MJ/kg/day or 23 kcal/kg/day
139 Steady state ρ M dm = I ɛm b = 0 M =(I b)/ɛ M 1 ɛ I ε decreases with weight and increases with activity ε ~ 0.1 MJ/kg/day or 23 kcal/kg/day Extra ~23 kcal/day is an extra kg
140 Time Constant ρ M dm = I ɛm b Time constant (half-life/.69) to reach steady state: τ = ρ M /ɛ
141 Time Constant ρ M dm = I ɛm b Time constant (half-life/.69) to reach steady state: τ = ρ M /ɛ ρm ~ 7700 kcal/kg, ε ~ 23 kcal/day, τ ~1 year
142 Time Constant ρ M dm = I ɛm b Time constant (half-life/.69) to reach steady state: τ = ρ M /ɛ ρm ~ 7700 kcal/kg, ε ~ 23 kcal/day, τ ~1 year τ increases with weight and decreases with activity
143 Intake precision To maintain weight to 2 kg requires controlling intake to ~50 kcal/day, (out of ~2500 kcal/day)
144 Intake precision To maintain weight to 2 kg requires controlling intake to ~50 kcal/day, (out of ~2500 kcal/day) Cookie is ~150 kcal
145 Intake precision To maintain weight to 2 kg requires controlling intake to ~50 kcal/day, (out of ~2500 kcal/day) Cookie is ~150 kcal Paradox?
146 Intake precision To maintain weight to 2 kg requires controlling intake to ~50 kcal/day, (out of ~2500 kcal/day) Cookie is ~150 kcal Paradox? No, because of long time constant
147 Example daily intake energy Beltsville one year intake study (courtesy of W. Rumpler) Intake Energy (kcal) Weight (kg) Days Days CV ~ 24% CV ~ 1% Intake varations have little effect on weight
148 Time varying intake I(t) =Ī + η(t) Noisy intake η(t)η(t ) = σ 2 δ(t t ) White noise ρ M dm = Ī b ɛm + η(t) Ornstein-Uhlenbeck process CV(I) is σ/i, find CV(M)
149 (M(t) M ) 2 = σ2 2ρ M ɛ M = I b ɛ CV(M) = 1 2τ Ī I b CV(Ī) τ = ρ M /ɛ
150 (M(t) M ) 2 = σ2 2ρ M ɛ M = I b ɛ CV(M) = 1 2τ Ī I b CV(Ī) τ = ρ M /ɛ
151 (M(t) M ) 2 = σ2 2ρ M ɛ M = I b ɛ CV(M) = 1 2τ Ī I b CV(Ī) τ = ρ M /ɛ For 2τ 30, Ī 2500,b 600
152 (M(t) M ) 2 = σ2 2ρ M ɛ M = I b ɛ CV(M) = 1 2τ Ī I b CV(Ī) τ = ρ M /ɛ For 2τ 30, Ī 2500,b 600 CV (M) reduced by factor of vs CV (I)
153 Simulated Beltsville data 10 years Intake Energy (kcal) Weight (kg) Days Days CV ~ 23% CV ~ 2%
154 Correlations increase fluctuations Intake Energy (kcal) Weight (kg) Days Days CV ~ 26% CV ~ 5%
155 Correlations increase fluctuations Intake Energy (kcal) Weight (kg) Days Days CV ~ 26% CV ~ 5% Longer correlations higher BMI Periwal and Chow, AJP:EM, 291: (2006)
156 Acknowledgments Kevin Hall Vipul Periwal Heather Bain Michael Dore Juen Guo
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